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MCD6110-心理代写
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MCD6110
Psychology 1B
Lectures 1 & 2, Week 8
Psychological Discovery:
Summarising Data Using
Descriptive Statistics
COMMONWEALTH OF AUSTRALIA
Copyright Regulations 1969
WARNING
This material has been reproduced and communicated to
you by or on behalf of Monash University pursuant to Part
VB of the Copyright Act 1968 (the Act).
The material in this communication may be subject to
copyright under the Act. Any further reproduction or
communication of this material by you may be the subject of
copyright protection under the Act.
Do not remove this notice.
Learning Outcomes W7, L1 & L2
1.Distinguish between descriptive and inferential statistics;
2.Organise and summarise data in the form of frequency distribution
tables and graphs;
3.Calculate and differentiate between the three measures of central
tendency (mean, median, and mode);
4.Describe and calculate measures of variability or dispersion (range,
variance, standard deviation);
5.Describe the position of a particular score in a distribution using z-
scores; and
6.Summarise the properties of the standard normal distribution and its
relationship with probability.
3
2. Form a
Hypothesis
3. Define and
Measure
Variables
4. Identify
Participants or
Subjects
5. Select a
Research
Strategy
6. Select a
Research
Design
7. Conduct the
Study
8. Evaluate the
Data
9. Report the
Results
10. Refine or
Reformulate
Research Idea
1. Find a
Research Idea
Research Methods for the Behavioral Sciences, Fourth
Edition by Frederick J Gravetter and Lori-Ann B. Forzano
Copyright © 2011 Wadsworth Publishing, a division of
Cengage Learning. All rights reserved.
Conducting the Study
▪ Recruit the participants
▪ Apply your measurement procedures
▪ Collect the data!
5
6Descriptive Statistics
▪ After we have collected our raw data, we need to reduce the endless list
of numbers into a manageable and meaningful summary. This is called
descriptive statistics.
▪ Descriptive statistics entail the organisation, summarisation, and
simplification of raw data so that patterns and trends in variables can
be seen
– i.e., making order out of chaos!
7Descriptive vs. Inferential Statistics
▪ Descriptive statistics are methods for simplifying, organizing and
summarizing data
– Note that a descriptive value for a population is called a
parameter (symbolised by Greek letters, μ, σ, σ2, etc.) and a
descriptive value for a sample is called a statistic (symbolised by
regular (English) letters, M, s, s2, etc.)
▪ Inferential statistics are methods for using sample data to make
general conclusions (inferences) about populations.
– i.e., sample statistics are used as the basis for drawing
conclusions about population parameters
▪ A frequency distribution is an organized tabulation showing exactly
how many individuals are located in each category on the scale of
measurement
▪ A frequency distribution presents an organized picture of the entire set
of scores, and it shows where each individual is located relative to
others in the distribution
– i.e., how did each individual score in relation to the rest of the
group
8
Frequency Distributions: Organising Data
9Frequency Distribution: Making Sense of Data
▪ A frequency distribution, specifies the number of times each score
occurred on the scale of measurement
▪ You can organise and summarise data using tabular or graphical
techniques but not both
Fig. 2-2, p. 45
Statistics for the Behavioral Sciences, Eighth Edition
by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a
division of Thomson Learning. All rights reserved.
10
Frequency Distribution Tables
▪ Consist of two columns - one listing categories
on the scale of measurement (X) and another
for frequency (Y).
▪ In X column list values from highest to lowest.
▪
▪ The frequency (f) column lists how many times
the same score is present
▪ The sum of the frequencies should equal N.
X f
5
4
3
2
1
1
2
3
3
1
Σf = 10
Number of
Words
Recalled
(X)
Frequency
(f)
12 1
11 2
10 3
9 4
8 3
7 2
6 1
∑f = 16
11
Activity 1: Frequency Distribution Table
▪ In your cognition lectures you were
given a list of 15 words to memorise,
and later asked to recall these
words. Data from 16 students is
provided below. Organise the data
into the table.
▪ Number of Words Recalled:
9, 8, 9, 10, 9, 11, 6, 8, 8, 10,
7, 9, 7, 11, 12, 10
12
Grouped Frequency Distribution Table
▪ When a set of scores covers a wide range of values. In these
situations, a list of all the X values would be too long
▪ To remedy this situation, a grouped frequency distribution table is
used.
13
▪ The X column lists groups of scores, called
class intervals.
▪ These intervals all have the same width.
▪ Each interval begins with a value that is a
multiple of the interval width.
▪ The interval width is selected so that the table
will have approximately ten intervals.
X f
90-94
85-89
80-84
75-79
70-74
65-69
60-64
55-59
50-54
3
4
5
4
3
1
3
1
1
Σf = 25
Exam scores:
82, 75, 88, 93, 53, 84, 87,
58, 72, 94, 69, 84, 61, 91,
64, 87, 84, 70, 76, 89, 75,
80, 73, 78, 60
14
Frequency Distribution Graphs
▪ In a frequency distribution graph, the score categories (X values)
are listed on the X axis and the frequencies are listed on the Y axis.
▪ When the score categories consist of numerical scores from an interval
or ratio scale, the graph should be either a histogram or a polygon.
Fig. 2-12, p. 51
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Histograms
In a histogram, a bar is
centered above each score
so that the height of the bar
corresponds to the
frequency and the width
extends to the real limits,
so that adjacent bars
touch.
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Polygons
Polygon, a dot is centered
above each score.
The height of the dot
corresponds to the frequency.
The dots are then connected by
straight lines.
A line is drawn at each end to
bring the graph back to zero f.
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Fig. 2-6, p. 47
17
Bar Graphs
▪ When the score categories (X values) are measurements from a
nominal or an ordinal scale, the graph should be a bar graph.
▪ A bar graph is just like a histogram except that gaps are left between
adjacent bars (because the data is discrete rather than continuous).
0
10
20
30
40
50
60
F
re
q
u
e
n
c
y
Words Recalled
18
Relative Frequency
▪ When populations are too large to
know exact number of individuals
use a relative frequency distribution.
each category.
Smooth Curve
▪ If the scores in the population are measured on an interval or ratio
scale, present the distribution as a smooth curve.
▪ The smooth curve emphasizes the fact that the distribution is not
showing the exact frequency for each category.
Statistics for the Behavioral Sciences, Eighth Edition by
Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of
Thomson Learning. All rights reserved.
Fig. 2-9, p. 48
20
Frequency Distribution Graphs: Summary
▪ Frequency distribution graphs are useful because they show the entire
set of scores
▪ At a glance, you can determine the highest score, the lowest score, and
where the scores are centered
▪ The graph also shows whether the scores are clustered together or
scattered over a wide range
▪ Frequency distributions tell us about the:
– Shape of the distribution (i.e., symmetrical or skewed)
– Central tendency (i.e., where do most scores fall)
– Variability (i.e., what is the spread of scores)
21
Shape
▪ A graph shows the shape of the distribution
▪ A distribution is symmetrical if the left side of the graph is (roughly) a
mirror image of the right side
▪ One example of a symmetrical distribution is the bell-shaped normal
distribution
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved. Fig. 2-11, p. 50
22
Skewed Distributions
▪ Distributions are skewed when scores pile up
on one side of the distribution, leaving a "tail"
of a few extreme values on the other side
▪ In a positively skewed distribution, the
scores tend to pile up on the left side of the
distribution with the tail tapering off to the right
▪ In a negatively skewed distribution, the
scores tend to pile up on the right side and the
tail points to the left
Fig. 2-11, p. 50
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
23
Central Tendency
▪ Central tendency determines a single value that accurately describes
the center of the distribution & represents entire distribution of scores
▪ The goal of central tendency is to identify the single value that is the
best representative for the entire set of data (i.e., the average value)
▪ By identifying the "average score," central tendency allows researchers
to summarize or condense a large set of data into a single value
▪ Central tendency serves as a descriptive statistic because it allows
researchers to describe a set of data in a very simple, concise form
24
Central Tendency and Variability
▪ “Central tendency measures
where the center of the distribution
is located” (Gravetter & Wallnau, 2009, p. 50)
▪ “Variability tells whether the
scores are spread over a wide
range or are clustered together”
(Gravetter & Wallnau, 2009, p. 50)
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved. Modified Fig. 2-11, p. 50
25
Measuring Central Tendency
▪ There are three different measures of central tendency:
– The Mean
– The Median
– The Mode
▪ The most appropriate measure of central tendency will be determined by:
– What scale of measurement you used (i.e., nominal, ordinal,
interval, or ratio)
– The shape of your frequency distribution (i.e., symmetrical or
skewed)
26
The Mean
▪ The mean is the most commonly used measure of central tendency
▪ It is the arithmetic average of the scores in a distribution
▪ Computation of the mean requires scores that are numerical values
measured on an interval or ratio scale
▪ The mean is obtained by computing the sum (ΣX), or total, for the
entire set of scores, then dividing this sum by the number of scores (N)
mean
Note that ത is also used to represent the mean
Calculating the Mean
▪ For populations:
(use Greek letters)
N
X
=
▪ For samples:
(use regular letters)
n
X
M
=
Population mean (μ)
Sample mean (M)
Sum of all X values
Sum of all X values
Sample size (n)
Population size (N)
28
When the Mean Won’t Work
– When a distribution contains a few extreme scores (or is very skewed),
the mean will be pulled toward the extremes (displaced toward the tail).
In this case, the mean will not provide a "central" value.
– For nominal and ordinal data
No skew
The Median
▪ The median is the score that divides a distribution exactly in half
– i.e., 50% of scores are equal
to or less than the median,
and 50% of scores are
above the median (for this
reason the median is also
referred to as the 50th
percentile)
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Fig. 3-6, p. 84
30
Calculating the Median
▪ If the scores in a distribution are listed in order from smallest to largest,
the median is defined as the midpoint of the list.
– Median can be used on an ordinal, interval, or ratio scale.
▪ The median can be found by a simple counting procedure:
– With an odd number of scores, list the values in order, and the
median is the middle score in the list.
– With an even number of scores, list the values in order, and the
median is half-way between the middle two scores.
31
Activity 3: Calculate the Median
▪ Number of words recalled by sample of 16 participants:
9, 8, 9, 10, 9, 11, 6, 8, 8, 10, 7, 9, 7, 11, 12, 10
Calculate the median (Mdn)
Re-list scores in order from
lowest to highest
=
+ 1
2
=
16 + 1
2
=
17
2
= 8.5ℎ
6, 7, 7, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11, 11, 12
Advantages of the Median
▪ It is relatively unaffected by extreme scores.
▪ The median tends to stay in the "center" of the distribution
even despite extreme scores or when the distribution is
very skewed.
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Fig. 3-14, p. 97
33
The Mode
▪ The mode is defined as the most frequently occurring
category or score in the distribution.
▪ In a frequency distribution graph, the mode is the category
or score corresponding to the peak or high point of the
distribution.
▪ The mode can be determined for data measured on any
scale of measurement: nominal, ordinal, interval, or ratio.
▪ The primary value of the mode is that it is the only measure
of central tendency that can be used for data measured on
a nominal scale.
34
Calculating the Mode
▪ Referring to a frequency distribution
table or graph , the mode is the
score or category with the highest
frequency/peak
Fig. 3-10, p. 91
Mode = 11
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B.
Wallnau Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All
rights reserved.
Central Tendency and the Shape of the
Distribution
▪ In a symmetrical distribution with
only one peak, the mean, median,
and mode will always be equal.
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Fig. 3-13, p. 96
36
▪ In a skewed distribution, the mode will be located at the peak on
one side and the mean usually will be displaced toward the tail
on the other side.
▪ The Median is usually located between the Mean and Mode.
Fig. 3-14, p. 97
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Learning Outcomes L2
1.Distinguish between descriptive and inferential statistics;
2.Organise and summarise data in the form of frequency distribution
tables and graphs;
3.Calculate and differentiate between the three measures of central
tendency (mean, median, and mode);
4.Describe and calculate measures of variability or dispersion (range,
variance, standard deviation);
5.Describe the position of a particular score in a distribution using z-
scores; and
6.Summarise the properties of the standard normal distribution and its
relationship with probability.
37
38
Central Tendency and Variability
▪ The goal for variability is to obtain a measure of how spread out the
scores are in a distribution
▪ Thus, central tendency describes the central point of the
distribution, and variability describes how the scores are scattered
around that central point
▪ Together, central tendency and variability are the two primary values
that are used to describe a distribution of scores
– i.e., a measure of variability usually accompanies a measure of
central tendency as basic descriptive statistics for a set of
scores
39
Variability
▪ Variability serves both as a descriptive measure and as an important
component of most inferential statistics
▪ As a descriptive statistic, variability measures the degree to which the
scores are spread out or clustered together in a distribution
▪ In the context of inferential statistics, variability provides a measure of
how accurately any individual score or sample represents the entire
population
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Fig. 4-1, p. 106
▪ When the variability is small, all of
the scores are clustered close
together and any individual score or
sample will necessarily provide a
good representation of the entire
set.
▪ On the other hand, when variability
is large and scores are widely
spread, it is easy for one or two
extreme scores to give a distorted
picture of the general population.
41
Measuring Variability
▪ Variability can be measured with
– the range
– the standard deviation / variance
▪ In each case, variability is determined by measuring distance, for
example:
– distance between lowest and highest scores (range)
– standard (i.e., average) distance between a score and the mean
(standard deviation)
• Note that variance = average squared distance between a
score and the mean
42
The Range
▪ The range is the total distance covered by the distribution
▪ . Range = highest score – lowest score
▪ It is completely determined by the most extreme values which leads to
two issues:
– An unusually large or small score will produce a large range even
if the other scores are closely clustered together
– Because only the smallest and largest scores are considered, it
does not provide an accurate representation of the distribution’s
variability. The Range is crude folks!
43
The Standard Deviation
▪ Standard Deviation measures the standard (i.e., average) distance
between a score and the mean
▪ In the above graph the arrows represent deviations – that is, how far each
score is from the mean
– Half the deviations will be positive (i.e., when people score higher
than the mean)
– Half the deviations will be negative (i.e., when people score lower
than the mean)
mean
+ve-ve
3. Compute the mean for the squared deviations – this is called
the variance
4. Take the square root of the variance – this is called the
standard deviation
44
Calculating a Standard Deviation
1. Calculate each score’s deviation (distance from mean)
2. Square each deviation (because otherwise the +ve and –ve
deviations will cancel out to zero)
mean
difference
between
each score
and mean
45
Activity 1: Calculate the Sample Standard Deviation
= − =
Sample variance (s2):
2 =
− 1
=
Sample standard deviation (s or SD):
=
= 2 =
40
16 − 1
=
40
15
= 2.666667
2.666667 = 1.63
40
X X – M (X – M)2
9
8
9
10
9
11
6
8
8
10
7
9
7
11
12
10
X X – (X – )2
9 0 0
8 -1 1
9 0 0
10 1 1
9 0 0
11 2 4
6 -3 9
8 -1 1
8 -1 1
10 1 1
7 -2 4
9 0 0
7 -2 4
11 2 4
12 3 9
10 1 1
46
Sample Standard Deviations
▪ We use the same formulae as for populations, but we use:
– M to represent the sample mean
– n to represent the sample size
– s2 to represent the sample variance
– s to represent the sample standard deviation
– And most importantly, we use ‘n – 1’ in our calculations of
variance instead of N
▪ Sample variability is a biased estimate of population variability – it
tends to underestimate the variability of scores in the population.
So we apply a correction of ‘n – 1’
Standard Deviation for the Population
▪ For the Population, the Standard Deviation is not calculated with n-1
▪ Because we are not mimicking the population-IT IS the population..
▪² =
4728th February 2011Pres ntation title
▪ If you have numerical values for the mean and the standard deviation,
you should be able to construct a visual image (or a sketch) of the
distribution of scores
48
Graphical Conclusions
49
z-Scores and Location
▪ By itself, a raw score or X value provides very little information about
how that particular score compares with other values in the distribution
▪ A score of X = 76, for example, may be a relatively low score, or an
average score, or an extremely high score depending on the mean and
standard deviation for the distribution from which the score was
obtained
▪ If the raw score is transformed into a z-score (i.e., a standardised
score), however, the value of the z-score tells you exactly where the
score is located relative to all the other scores in the distribution
50
z-Scores
▪ The process of changing an X value into a z-score involves creating a
signed number, called a z-score, such that
a) The sign of the z-score (+ or –) identifies whether the X value is
located above the mean (positive) or below the mean
(negative)
b) The numerical value of the z-score corresponds to the number
of standard deviations between X and the mean of the
distribution
c) A z-score of zero implies that the X value is exactly equal to
the mean
Graphic Representation of Location
▪ A z-score of +1 indicates that the score is located one standard
deviation (1σ) above the mean (μ)
▪ A z-score of -2 indicates that the score is located two standard
deviations (2σ) below the mean (μ)
Fig. 5-3, p. 141
Statistics for the Behavioral Sciences, Eighth Edition
by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a
division of Thomson Learning. All rights reserved.
52
Transforming X values into z-Scores
▪ For populations:
▪ For samples:
=
−
=
−
X value (raw score)
Population mean
Population standard deviation
X value (raw score)
Sample mean
Sample standard deviation
53
Activity 2: Calculate the Z-Score
The Weschler Memory Scale (WMS-IV) can be used to measure several
facets of memory, including verbal-auditory memory. The Auditory
Memory Index has a population mean (µ) of 100, and population standard
deviation (σ) of 15.
You test the memory of a sufferer of schizophrenia because you suspect
that the disorganised thinking and auditory hallucinations which are
characteristic of this disorder may interfere with auditory memory. The
person receives a score (X) of 80 on the Auditory Memory Index of the
WMS-IV.
=
−
=
−
=
−
= −.
54
Transforming z-scores into X-values
▪ The terms in the formula can be rearranged to create an equation for
computing the value of X corresponding to any specific z-score
– X = µ + z(σ) OR X = M + z(s)
Example: For IQ scores μ = 100 and σ = 15. To join MENSA I need a z-
score of at least +2. What is the corresponding IQ score for z = +2?
zX +=
( )152100 +=
30100+=
130=
55
z-Scores as a Standardised Distribution
▪ When an entire distribution of X values is transformed into z-scores, the
resulting distribution of z-scores will always have a mean of zero and a
standard deviation of one
▪ The transformation does not change the shape of the original
distribution and it does not change the location of any individual score
relative to others in the distribution
▪ Transforming an entire distribution basically just involves re-labeling the
X-axis in z-score units
Fig. 5-5, p. 146
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
An entire population of scores is transformed into z-scores. The
transformation does not change the shape of the distribution, but the
mean is transformed from 100 into a value of 0, and the standard
deviation is transformed from 10 into a value of 1.
Transforming into a Standardised Distribution
56
57
Fig. 5-6, p. 147
Following a z-score transformation, the X-axis is relabeled in z-score units.
The distance that is equivalent to 1 standard deviation on the X-axis (σ = 10
points in this example) corresponds to 1 point on the z-score scale.
Statistics for the Behavioral Sciences, Eighth Edition
by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a
division of Thomson Learning. All rights reserved.
58
z-Scores as a Standardised Distribution
▪ The advantage of standardising distributions is that two (or more) different
distributions can be made the same
– For example, one distribution has μ = 100 and σ = 10, and another
distribution has μ = 40 and σ = 6
– When these distribution are transformed to z-scores, both will have μ =
0 and σ = 1
▪ Because z-score distributions all have the same mean and standard
deviation, individual scores from different distributions can be directly
compared.
– A z-score of +1.00 specifies the same location in all z-score
distributions
Fig. 5-2, p. 140
EXAM A
EXAM B
Mean, μ = 70
Standard deviation, σ = 3
Exam score, X = 76
2
3
7076
+=
−
=z
μ = 70, σ = 12, X = 76
5.0
12
7076
+=
−
=z
Statistics for the Behavioral Sciences, Eighth Edition
by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a
division of Thomson Learning. All rights reserved.
Frequency Distribution Graphs and Probability
▪ When a population of scores is
represented by a frequency
distribution, probabilities can be
defined by proportions of the
distribution
▪ In graphs, probability can be defined
as a proportion of area under the
curve
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Fig. 6-2, p. 169
130
Fig. 6-4, p. 171
The Standard Normal Distribution
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Transforming all raw scores into z-
scores gives rise to the Standard
Normal Distribution which has:
mean (μ) = 0, and
standard deviation (σ) = 1
~ 68%
~ 95%
61
62
▪ The ability to identify the area under the curve corresponding to
particular z-scores, makes the standard normal distribution of particular
usefulness for inferential statistics
▪ For example, if a normal distribution is transformed into the standard
normal distribution, we know that approximately:
– 68% of values will lie within 1 SD of the mean
(i.e., between the z-scores of -1 and +1);
– 95% of values will lie within 2 SDs of the mean
(i.e., between the z-scores of -2 and +2); and
– 99.7% of values will lie within 3 SDs of the mean
(i.e., between the z-scores of -3 and +3)
63
The Unit Normal Table
▪ The unit normal table (see textbook appendix) lists several different
proportions corresponding to each z-score location:
– Column 1 of the table lists z-score values.
– Finally, column 2 lists the proportion between the mean and the z-
score location.
– Finally, for each z-score location, column 3 lists the proportions in
the beyond z (i.e., further out in the tail).
▪ Because probability equals proportion multiplied by 100, the table
values can also be used to determine probabilities
– Note that because the distribution is symmetrical, the proportions
on the right hand side are the same as the proportions on the left
hand side
64
65
Probability and the Standard Normal Distribution
▪ To find the probability corresponding to a particular score (X value):
1. Transform the score into a z-score,
2. Look up the z-score in the table,
3. Read across the row to find the appropriate proportion (multiply by
100 to convert to probability).
▪ To find the score (X value) corresponding to a particular proportion:
1. Look up the proportion in the table,
2. Read across the row to find the corresponding z-score,
3. Transform the z-score into an X value.
66
Activity 3: Calculate the Probability
The Auditory Memory Index of the WMS-IV has a population
mean (µ) of 100, and population standard deviation (σ) of 15.
A sufferer of schizophrenia receives a score (X) of 80.
What is:
▪ the probability of receiving a score lower than 80
< 80 = < −1.33 =
▪ the probability of receiving a score higher than 80
> 80 = > −1.33 =
67
< 80 = < −1.33
= 0.09175 or 9.2%
9.175%
68
> 80 = > −1.33
= 1 − < −1.33
= 1 − 0.09175
= 0.90825 or 90.8%
9.175%
90.875%
69
▪ ANY QUESTIONS?
▪ ………….Thank you
Psychology 1B
Lectures 1 & 2, Week 8
Psychological Discovery:
Summarising Data Using
Descriptive Statistics
COMMONWEALTH OF AUSTRALIA
Copyright Regulations 1969
WARNING
This material has been reproduced and communicated to
you by or on behalf of Monash University pursuant to Part
VB of the Copyright Act 1968 (the Act).
The material in this communication may be subject to
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Learning Outcomes W7, L1 & L2
1.Distinguish between descriptive and inferential statistics;
2.Organise and summarise data in the form of frequency distribution
tables and graphs;
3.Calculate and differentiate between the three measures of central
tendency (mean, median, and mode);
4.Describe and calculate measures of variability or dispersion (range,
variance, standard deviation);
5.Describe the position of a particular score in a distribution using z-
scores; and
6.Summarise the properties of the standard normal distribution and its
relationship with probability.
3
2. Form a
Hypothesis
3. Define and
Measure
Variables
4. Identify
Participants or
Subjects
5. Select a
Research
Strategy
6. Select a
Research
Design
7. Conduct the
Study
8. Evaluate the
Data
9. Report the
Results
10. Refine or
Reformulate
Research Idea
1. Find a
Research Idea
Research Methods for the Behavioral Sciences, Fourth
Edition by Frederick J Gravetter and Lori-Ann B. Forzano
Copyright © 2011 Wadsworth Publishing, a division of
Cengage Learning. All rights reserved.
Conducting the Study
▪ Recruit the participants
▪ Apply your measurement procedures
▪ Collect the data!
5
6Descriptive Statistics
▪ After we have collected our raw data, we need to reduce the endless list
of numbers into a manageable and meaningful summary. This is called
descriptive statistics.
▪ Descriptive statistics entail the organisation, summarisation, and
simplification of raw data so that patterns and trends in variables can
be seen
– i.e., making order out of chaos!
7Descriptive vs. Inferential Statistics
▪ Descriptive statistics are methods for simplifying, organizing and
summarizing data
– Note that a descriptive value for a population is called a
parameter (symbolised by Greek letters, μ, σ, σ2, etc.) and a
descriptive value for a sample is called a statistic (symbolised by
regular (English) letters, M, s, s2, etc.)
▪ Inferential statistics are methods for using sample data to make
general conclusions (inferences) about populations.
– i.e., sample statistics are used as the basis for drawing
conclusions about population parameters
▪ A frequency distribution is an organized tabulation showing exactly
how many individuals are located in each category on the scale of
measurement
▪ A frequency distribution presents an organized picture of the entire set
of scores, and it shows where each individual is located relative to
others in the distribution
– i.e., how did each individual score in relation to the rest of the
group
8
Frequency Distributions: Organising Data
9Frequency Distribution: Making Sense of Data
▪ A frequency distribution, specifies the number of times each score
occurred on the scale of measurement
▪ You can organise and summarise data using tabular or graphical
techniques but not both
Fig. 2-2, p. 45
Statistics for the Behavioral Sciences, Eighth Edition
by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a
division of Thomson Learning. All rights reserved.
10
Frequency Distribution Tables
▪ Consist of two columns - one listing categories
on the scale of measurement (X) and another
for frequency (Y).
▪ In X column list values from highest to lowest.
▪
▪ The frequency (f) column lists how many times
the same score is present
▪ The sum of the frequencies should equal N.
X f
5
4
3
2
1
1
2
3
3
1
Σf = 10
Number of
Words
Recalled
(X)
Frequency
(f)
12 1
11 2
10 3
9 4
8 3
7 2
6 1
∑f = 16
11
Activity 1: Frequency Distribution Table
▪ In your cognition lectures you were
given a list of 15 words to memorise,
and later asked to recall these
words. Data from 16 students is
provided below. Organise the data
into the table.
▪ Number of Words Recalled:
9, 8, 9, 10, 9, 11, 6, 8, 8, 10,
7, 9, 7, 11, 12, 10
12
Grouped Frequency Distribution Table
▪ When a set of scores covers a wide range of values. In these
situations, a list of all the X values would be too long
▪ To remedy this situation, a grouped frequency distribution table is
used.
13
▪ The X column lists groups of scores, called
class intervals.
▪ These intervals all have the same width.
▪ Each interval begins with a value that is a
multiple of the interval width.
▪ The interval width is selected so that the table
will have approximately ten intervals.
X f
90-94
85-89
80-84
75-79
70-74
65-69
60-64
55-59
50-54
3
4
5
4
3
1
3
1
1
Σf = 25
Exam scores:
82, 75, 88, 93, 53, 84, 87,
58, 72, 94, 69, 84, 61, 91,
64, 87, 84, 70, 76, 89, 75,
80, 73, 78, 60
14
Frequency Distribution Graphs
▪ In a frequency distribution graph, the score categories (X values)
are listed on the X axis and the frequencies are listed on the Y axis.
▪ When the score categories consist of numerical scores from an interval
or ratio scale, the graph should be either a histogram or a polygon.
Fig. 2-12, p. 51
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Histograms
In a histogram, a bar is
centered above each score
so that the height of the bar
corresponds to the
frequency and the width
extends to the real limits,
so that adjacent bars
touch.
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Polygons
Polygon, a dot is centered
above each score.
The height of the dot
corresponds to the frequency.
The dots are then connected by
straight lines.
A line is drawn at each end to
bring the graph back to zero f.
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Fig. 2-6, p. 47
17
Bar Graphs
▪ When the score categories (X values) are measurements from a
nominal or an ordinal scale, the graph should be a bar graph.
▪ A bar graph is just like a histogram except that gaps are left between
adjacent bars (because the data is discrete rather than continuous).
0
10
20
30
40
50
60
F
re
q
u
e
n
c
y
Words Recalled
18
Relative Frequency
▪ When populations are too large to
know exact number of individuals
use a relative frequency distribution.
each category.
Smooth Curve
▪ If the scores in the population are measured on an interval or ratio
scale, present the distribution as a smooth curve.
▪ The smooth curve emphasizes the fact that the distribution is not
showing the exact frequency for each category.
Statistics for the Behavioral Sciences, Eighth Edition by
Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of
Thomson Learning. All rights reserved.
Fig. 2-9, p. 48
20
Frequency Distribution Graphs: Summary
▪ Frequency distribution graphs are useful because they show the entire
set of scores
▪ At a glance, you can determine the highest score, the lowest score, and
where the scores are centered
▪ The graph also shows whether the scores are clustered together or
scattered over a wide range
▪ Frequency distributions tell us about the:
– Shape of the distribution (i.e., symmetrical or skewed)
– Central tendency (i.e., where do most scores fall)
– Variability (i.e., what is the spread of scores)
21
Shape
▪ A graph shows the shape of the distribution
▪ A distribution is symmetrical if the left side of the graph is (roughly) a
mirror image of the right side
▪ One example of a symmetrical distribution is the bell-shaped normal
distribution
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved. Fig. 2-11, p. 50
22
Skewed Distributions
▪ Distributions are skewed when scores pile up
on one side of the distribution, leaving a "tail"
of a few extreme values on the other side
▪ In a positively skewed distribution, the
scores tend to pile up on the left side of the
distribution with the tail tapering off to the right
▪ In a negatively skewed distribution, the
scores tend to pile up on the right side and the
tail points to the left
Fig. 2-11, p. 50
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
23
Central Tendency
▪ Central tendency determines a single value that accurately describes
the center of the distribution & represents entire distribution of scores
▪ The goal of central tendency is to identify the single value that is the
best representative for the entire set of data (i.e., the average value)
▪ By identifying the "average score," central tendency allows researchers
to summarize or condense a large set of data into a single value
▪ Central tendency serves as a descriptive statistic because it allows
researchers to describe a set of data in a very simple, concise form
24
Central Tendency and Variability
▪ “Central tendency measures
where the center of the distribution
is located” (Gravetter & Wallnau, 2009, p. 50)
▪ “Variability tells whether the
scores are spread over a wide
range or are clustered together”
(Gravetter & Wallnau, 2009, p. 50)
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved. Modified Fig. 2-11, p. 50
25
Measuring Central Tendency
▪ There are three different measures of central tendency:
– The Mean
– The Median
– The Mode
▪ The most appropriate measure of central tendency will be determined by:
– What scale of measurement you used (i.e., nominal, ordinal,
interval, or ratio)
– The shape of your frequency distribution (i.e., symmetrical or
skewed)
26
The Mean
▪ The mean is the most commonly used measure of central tendency
▪ It is the arithmetic average of the scores in a distribution
▪ Computation of the mean requires scores that are numerical values
measured on an interval or ratio scale
▪ The mean is obtained by computing the sum (ΣX), or total, for the
entire set of scores, then dividing this sum by the number of scores (N)
mean
Note that ത is also used to represent the mean
Calculating the Mean
▪ For populations:
(use Greek letters)
N
X
=
▪ For samples:
(use regular letters)
n
X
M
=
Population mean (μ)
Sample mean (M)
Sum of all X values
Sum of all X values
Sample size (n)
Population size (N)
28
When the Mean Won’t Work
– When a distribution contains a few extreme scores (or is very skewed),
the mean will be pulled toward the extremes (displaced toward the tail).
In this case, the mean will not provide a "central" value.
– For nominal and ordinal data
No skew
The Median
▪ The median is the score that divides a distribution exactly in half
– i.e., 50% of scores are equal
to or less than the median,
and 50% of scores are
above the median (for this
reason the median is also
referred to as the 50th
percentile)
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Fig. 3-6, p. 84
30
Calculating the Median
▪ If the scores in a distribution are listed in order from smallest to largest,
the median is defined as the midpoint of the list.
– Median can be used on an ordinal, interval, or ratio scale.
▪ The median can be found by a simple counting procedure:
– With an odd number of scores, list the values in order, and the
median is the middle score in the list.
– With an even number of scores, list the values in order, and the
median is half-way between the middle two scores.
31
Activity 3: Calculate the Median
▪ Number of words recalled by sample of 16 participants:
9, 8, 9, 10, 9, 11, 6, 8, 8, 10, 7, 9, 7, 11, 12, 10
Calculate the median (Mdn)
Re-list scores in order from
lowest to highest
=
+ 1
2
=
16 + 1
2
=
17
2
= 8.5ℎ
6, 7, 7, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11, 11, 12
Advantages of the Median
▪ It is relatively unaffected by extreme scores.
▪ The median tends to stay in the "center" of the distribution
even despite extreme scores or when the distribution is
very skewed.
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Fig. 3-14, p. 97
33
The Mode
▪ The mode is defined as the most frequently occurring
category or score in the distribution.
▪ In a frequency distribution graph, the mode is the category
or score corresponding to the peak or high point of the
distribution.
▪ The mode can be determined for data measured on any
scale of measurement: nominal, ordinal, interval, or ratio.
▪ The primary value of the mode is that it is the only measure
of central tendency that can be used for data measured on
a nominal scale.
34
Calculating the Mode
▪ Referring to a frequency distribution
table or graph , the mode is the
score or category with the highest
frequency/peak
Fig. 3-10, p. 91
Mode = 11
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B.
Wallnau Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All
rights reserved.
Central Tendency and the Shape of the
Distribution
▪ In a symmetrical distribution with
only one peak, the mean, median,
and mode will always be equal.
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Fig. 3-13, p. 96
36
▪ In a skewed distribution, the mode will be located at the peak on
one side and the mean usually will be displaced toward the tail
on the other side.
▪ The Median is usually located between the Mean and Mode.
Fig. 3-14, p. 97
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Learning Outcomes L2
1.Distinguish between descriptive and inferential statistics;
2.Organise and summarise data in the form of frequency distribution
tables and graphs;
3.Calculate and differentiate between the three measures of central
tendency (mean, median, and mode);
4.Describe and calculate measures of variability or dispersion (range,
variance, standard deviation);
5.Describe the position of a particular score in a distribution using z-
scores; and
6.Summarise the properties of the standard normal distribution and its
relationship with probability.
37
38
Central Tendency and Variability
▪ The goal for variability is to obtain a measure of how spread out the
scores are in a distribution
▪ Thus, central tendency describes the central point of the
distribution, and variability describes how the scores are scattered
around that central point
▪ Together, central tendency and variability are the two primary values
that are used to describe a distribution of scores
– i.e., a measure of variability usually accompanies a measure of
central tendency as basic descriptive statistics for a set of
scores
39
Variability
▪ Variability serves both as a descriptive measure and as an important
component of most inferential statistics
▪ As a descriptive statistic, variability measures the degree to which the
scores are spread out or clustered together in a distribution
▪ In the context of inferential statistics, variability provides a measure of
how accurately any individual score or sample represents the entire
population
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Fig. 4-1, p. 106
▪ When the variability is small, all of
the scores are clustered close
together and any individual score or
sample will necessarily provide a
good representation of the entire
set.
▪ On the other hand, when variability
is large and scores are widely
spread, it is easy for one or two
extreme scores to give a distorted
picture of the general population.
41
Measuring Variability
▪ Variability can be measured with
– the range
– the standard deviation / variance
▪ In each case, variability is determined by measuring distance, for
example:
– distance between lowest and highest scores (range)
– standard (i.e., average) distance between a score and the mean
(standard deviation)
• Note that variance = average squared distance between a
score and the mean
42
The Range
▪ The range is the total distance covered by the distribution
▪ . Range = highest score – lowest score
▪ It is completely determined by the most extreme values which leads to
two issues:
– An unusually large or small score will produce a large range even
if the other scores are closely clustered together
– Because only the smallest and largest scores are considered, it
does not provide an accurate representation of the distribution’s
variability. The Range is crude folks!
43
The Standard Deviation
▪ Standard Deviation measures the standard (i.e., average) distance
between a score and the mean
▪ In the above graph the arrows represent deviations – that is, how far each
score is from the mean
– Half the deviations will be positive (i.e., when people score higher
than the mean)
– Half the deviations will be negative (i.e., when people score lower
than the mean)
mean
+ve-ve
3. Compute the mean for the squared deviations – this is called
the variance
4. Take the square root of the variance – this is called the
standard deviation
44
Calculating a Standard Deviation
1. Calculate each score’s deviation (distance from mean)
2. Square each deviation (because otherwise the +ve and –ve
deviations will cancel out to zero)
mean
difference
between
each score
and mean
45
Activity 1: Calculate the Sample Standard Deviation
= − =
Sample variance (s2):
2 =
− 1
=
Sample standard deviation (s or SD):
=
= 2 =
40
16 − 1
=
40
15
= 2.666667
2.666667 = 1.63
40
X X – M (X – M)2
9
8
9
10
9
11
6
8
8
10
7
9
7
11
12
10
X X – (X – )2
9 0 0
8 -1 1
9 0 0
10 1 1
9 0 0
11 2 4
6 -3 9
8 -1 1
8 -1 1
10 1 1
7 -2 4
9 0 0
7 -2 4
11 2 4
12 3 9
10 1 1
46
Sample Standard Deviations
▪ We use the same formulae as for populations, but we use:
– M to represent the sample mean
– n to represent the sample size
– s2 to represent the sample variance
– s to represent the sample standard deviation
– And most importantly, we use ‘n – 1’ in our calculations of
variance instead of N
▪ Sample variability is a biased estimate of population variability – it
tends to underestimate the variability of scores in the population.
So we apply a correction of ‘n – 1’
Standard Deviation for the Population
▪ For the Population, the Standard Deviation is not calculated with n-1
▪ Because we are not mimicking the population-IT IS the population..
▪² =
4728th February 2011Pres ntation title
▪ If you have numerical values for the mean and the standard deviation,
you should be able to construct a visual image (or a sketch) of the
distribution of scores
48
Graphical Conclusions
49
z-Scores and Location
▪ By itself, a raw score or X value provides very little information about
how that particular score compares with other values in the distribution
▪ A score of X = 76, for example, may be a relatively low score, or an
average score, or an extremely high score depending on the mean and
standard deviation for the distribution from which the score was
obtained
▪ If the raw score is transformed into a z-score (i.e., a standardised
score), however, the value of the z-score tells you exactly where the
score is located relative to all the other scores in the distribution
50
z-Scores
▪ The process of changing an X value into a z-score involves creating a
signed number, called a z-score, such that
a) The sign of the z-score (+ or –) identifies whether the X value is
located above the mean (positive) or below the mean
(negative)
b) The numerical value of the z-score corresponds to the number
of standard deviations between X and the mean of the
distribution
c) A z-score of zero implies that the X value is exactly equal to
the mean
Graphic Representation of Location
▪ A z-score of +1 indicates that the score is located one standard
deviation (1σ) above the mean (μ)
▪ A z-score of -2 indicates that the score is located two standard
deviations (2σ) below the mean (μ)
Fig. 5-3, p. 141
Statistics for the Behavioral Sciences, Eighth Edition
by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a
division of Thomson Learning. All rights reserved.
52
Transforming X values into z-Scores
▪ For populations:
▪ For samples:
=
−
=
−
X value (raw score)
Population mean
Population standard deviation
X value (raw score)
Sample mean
Sample standard deviation
53
Activity 2: Calculate the Z-Score
The Weschler Memory Scale (WMS-IV) can be used to measure several
facets of memory, including verbal-auditory memory. The Auditory
Memory Index has a population mean (µ) of 100, and population standard
deviation (σ) of 15.
You test the memory of a sufferer of schizophrenia because you suspect
that the disorganised thinking and auditory hallucinations which are
characteristic of this disorder may interfere with auditory memory. The
person receives a score (X) of 80 on the Auditory Memory Index of the
WMS-IV.
=
−
=
−
=
−
= −.
54
Transforming z-scores into X-values
▪ The terms in the formula can be rearranged to create an equation for
computing the value of X corresponding to any specific z-score
– X = µ + z(σ) OR X = M + z(s)
Example: For IQ scores μ = 100 and σ = 15. To join MENSA I need a z-
score of at least +2. What is the corresponding IQ score for z = +2?
zX +=
( )152100 +=
30100+=
130=
55
z-Scores as a Standardised Distribution
▪ When an entire distribution of X values is transformed into z-scores, the
resulting distribution of z-scores will always have a mean of zero and a
standard deviation of one
▪ The transformation does not change the shape of the original
distribution and it does not change the location of any individual score
relative to others in the distribution
▪ Transforming an entire distribution basically just involves re-labeling the
X-axis in z-score units
Fig. 5-5, p. 146
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
An entire population of scores is transformed into z-scores. The
transformation does not change the shape of the distribution, but the
mean is transformed from 100 into a value of 0, and the standard
deviation is transformed from 10 into a value of 1.
Transforming into a Standardised Distribution
56
57
Fig. 5-6, p. 147
Following a z-score transformation, the X-axis is relabeled in z-score units.
The distance that is equivalent to 1 standard deviation on the X-axis (σ = 10
points in this example) corresponds to 1 point on the z-score scale.
Statistics for the Behavioral Sciences, Eighth Edition
by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a
division of Thomson Learning. All rights reserved.
58
z-Scores as a Standardised Distribution
▪ The advantage of standardising distributions is that two (or more) different
distributions can be made the same
– For example, one distribution has μ = 100 and σ = 10, and another
distribution has μ = 40 and σ = 6
– When these distribution are transformed to z-scores, both will have μ =
0 and σ = 1
▪ Because z-score distributions all have the same mean and standard
deviation, individual scores from different distributions can be directly
compared.
– A z-score of +1.00 specifies the same location in all z-score
distributions
Fig. 5-2, p. 140
EXAM A
EXAM B
Mean, μ = 70
Standard deviation, σ = 3
Exam score, X = 76
2
3
7076
+=
−
=z
μ = 70, σ = 12, X = 76
5.0
12
7076
+=
−
=z
Statistics for the Behavioral Sciences, Eighth Edition
by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a
division of Thomson Learning. All rights reserved.
Frequency Distribution Graphs and Probability
▪ When a population of scores is
represented by a frequency
distribution, probabilities can be
defined by proportions of the
distribution
▪ In graphs, probability can be defined
as a proportion of area under the
curve
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Fig. 6-2, p. 169
130
Fig. 6-4, p. 171
The Standard Normal Distribution
Statistics for the Behavioral Sciences, Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
Copyright © 2009 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Transforming all raw scores into z-
scores gives rise to the Standard
Normal Distribution which has:
mean (μ) = 0, and
standard deviation (σ) = 1
~ 68%
~ 95%
61
62
▪ The ability to identify the area under the curve corresponding to
particular z-scores, makes the standard normal distribution of particular
usefulness for inferential statistics
▪ For example, if a normal distribution is transformed into the standard
normal distribution, we know that approximately:
– 68% of values will lie within 1 SD of the mean
(i.e., between the z-scores of -1 and +1);
– 95% of values will lie within 2 SDs of the mean
(i.e., between the z-scores of -2 and +2); and
– 99.7% of values will lie within 3 SDs of the mean
(i.e., between the z-scores of -3 and +3)
63
The Unit Normal Table
▪ The unit normal table (see textbook appendix) lists several different
proportions corresponding to each z-score location:
– Column 1 of the table lists z-score values.
– Finally, column 2 lists the proportion between the mean and the z-
score location.
– Finally, for each z-score location, column 3 lists the proportions in
the beyond z (i.e., further out in the tail).
▪ Because probability equals proportion multiplied by 100, the table
values can also be used to determine probabilities
– Note that because the distribution is symmetrical, the proportions
on the right hand side are the same as the proportions on the left
hand side
64
65
Probability and the Standard Normal Distribution
▪ To find the probability corresponding to a particular score (X value):
1. Transform the score into a z-score,
2. Look up the z-score in the table,
3. Read across the row to find the appropriate proportion (multiply by
100 to convert to probability).
▪ To find the score (X value) corresponding to a particular proportion:
1. Look up the proportion in the table,
2. Read across the row to find the corresponding z-score,
3. Transform the z-score into an X value.
66
Activity 3: Calculate the Probability
The Auditory Memory Index of the WMS-IV has a population
mean (µ) of 100, and population standard deviation (σ) of 15.
A sufferer of schizophrenia receives a score (X) of 80.
What is:
▪ the probability of receiving a score lower than 80
< 80 = < −1.33 =
▪ the probability of receiving a score higher than 80
> 80 = > −1.33 =
67
< 80 = < −1.33
= 0.09175 or 9.2%
9.175%
68
> 80 = > −1.33
= 1 − < −1.33
= 1 − 0.09175
= 0.90825 or 90.8%
9.175%
90.875%
69
▪ ANY QUESTIONS?
▪ ………….Thank you
