Okun
PSY 230
STUDY GUIDE #2
Graphs, Central Tendency, and the Shapes of Distributions
I.
Graphs
1. When is it appropriate to use a histogram? How is a histogram constructed and how is
it interpreted?
A histogram is appropriate for either unordered or ordered qualitative variables. The categories
of the variable are placed along the X axis. Then the frequencies associated with the categories
of the variable are listed on the Y axis. With an unordered qualitative variable, the order of the
categories on the X axis is completely arbitrary. In contrast, with an ordered qualitative variable,
the categories on the X axis should be ordered in either an ascending or descending manner.
Bars are drawn over each category of the variable that corresponds to its frequency of
occurrence.
Karate Belt
F
White
Yellow
Brown
Black
40
20
10
5
1
2
2. When is it appropriate to use a frequency polygon? How is a frequency polygon
constructed and how is it interpreted?
A frequency polygon is appropriate for quantitative variables. The values of the variable are
placed along the X axis. Then the frequencies associated with the values of the variable are
listed on the Y axis. Dots are drawn over each value of the variable that corresponds to its
frequency of occurrence. Then the dots are connected.
FREQUENCY DISTRIBUTION FOR NUMBER OF NIGHTS DRINKING DURING A
TYPICAL WEEK FOR 30 COLLEGE STUDENTS
X
F
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
2
3
4
3. When is it appropriate to use a stem-and-leaf display? How is a stem-and-leaf display
constructed and how is it interpreted?
A stem-and-leaf display is appropriate for quantitative variables. We will compute stem-andleaf displays only for ungrouped (raw) data. Each score is divided into two parts--a stem and a
leaf.
Score
f
100
99
98
96
95
94
93
92
91
90
89
88
87
86
85
84
82
81
79
77
76
60
1
1
4
3
2
2
5
2
3
1
4
5
1
4
1
3
1
2
2
1
1
1
STEM-AND-LEAF DISPLAY OF FIRST TEST
SCORES IN PSY 230 (N = 50)
10
9
8
7
6
0
01112233333445566688889
112444566667888889999
6799
0
4. What are the advantages of the stem-and-leaf display?
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[1]
[23]
[21]
[4]
[1]
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II. Central Tendency
6. What does central tendency refer to? What is the advantage of using an index of central
tendency as compared to a frequency distribution or a graph?
7. What does the mean refer to? Which type of variable is it
used with? How is the mean computed?
The mean for a distribution of scores is the arithmetic average.
It is obtained by summing all of the scores and dividing by the
sample size (N). It is used with quantitative variables.
For a sample, M = X/N; for a population, x = X/N
Consider the following sample of 13 quiz scores
5, 6, 8, 9, 10, 11, 11, 11, 12, 13, 14, 16, and 17
M = 5 + 6 + 8 + 9 + 10 + 11 + 11 + 11 + 12 + 13 + 14 + 16 + 17
__________________________________________________________
13
M = 143/13 = 11
Boys
5
8
9
11
11
12
16
Girls
6
10
11
13
14
17
X=72
X=71
M = 10.29 M = 11.83
7
8. What is the special property of the mean?
Given the sample mean and N-1 deviation scores in a
distribution, how can the remaining raw score be determined?
A special property of the mean is that, when the mean is
subtracted from each of the raw scores, the sum of these
deviation scores always equals zero
City
Temp.
Deviation Score (X - M)
Boise
5
5-20 = -15
Chicago
10
10-20 = -10
Detroit
15
15-20 = -5
Pittsburgh
22
22-20 = +2
Phoenix
48
48-20 = +28
______________________________________________________
 = 0
(X - M)= 0.
When we subtract the mean from a raw score (X - M), we get
a deviation score. The sign of the deviation score tells us
whether a score is above or below the mean. The value of the
deviation score tells us the distance of the score from the
mean.
9. Which type of variable is it typically used with? What does
the median refer to? How is it computed?
The median is used with qualitative–ordered variables and
sometimes with quantitative variables. For quantitative
variables, the median is a number that divides a distribution in
half when scores are put into an array (i.e., ordered from
lowest to highest). The median is equivalent to the 50th
percentile.
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How to Find the Median
Here are the steps involved in finding the median:
1.
Order the observations from lowest to highest(i.e.,
form an array).
2.
Count up the number of observations (N).
3.
Divide [N + 1] by 2 to identify the distance (i.e.,
how many observations) you must “walk” into the array
to locate the score that represents the median.
4A. If N is an odd number, proceed with your “walk” and
identify the score in the middle of the array. This
score is the median.
4B. If N is an even number, use your “walk” to identify
the two scores closest to the middle of the array.
Then compute the median by summing the two closest
scores and dividing by 2.
Computing the Median from Arrayed Distributions of Scores
17
16
14
13
12
11
11
11
10
9
8
6
5
10.
16
14
13
12
12
12
11
10
9
8
6
5
What does the mode refer to? Which type of variable is it
typically used with? How is it determined?
The mode is the score (or category) in a distribution that
occurs most frequently.
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III.
Shapes of distributions
11.
What is the difference between symmetrical and skewed
distributions? How are scores distributed in
(1) a normal distribution;(2) a positively skewed
distribution; and (3) a negatively skewed distribution?
In a symmetrical distribution, the right-hand side of a of
the distribution always will be a mirror image of the left-hand
side of the distribution. An example of a symmetrical
distribution is the normal curve.
*Normal:
A distribution in which scores are bunched up in
the middle and scores gradually become sparser at
the extremes. (See p. 11.)
In contrast to a normal distribution, sometimes
distributions are skewed. In a skewed distribution, most scores
are bunched up at one end of the distribution and scores occur
relatively infrequently at the other end of the distribution.
Skewed distributions are, by definition, asymmetrical.
*positive skew: A lopsided distribution in which most
scores are bunched up at the low end and only a few scores are
at the high end. Positively skewed distributions are said to be
skewed to the right because the tail goes to the right. (See p.
11.)
*negative skew: A lopsided distribution in which most
scores are bunched up at the high end and only a few scores are
at the low end. Negatively skewed distributions are said to be
skewed to the left because the tail goes to the left. (See p.
11.)
12.
What is a bimodal distribution? What does a bimodal
distribution indicate?
A bimodal distribution has two humps or peaks. Distributions
tend to be bimodal when the sample consists of two distinct
groups. For example, the weight of adults tends to have a
bimodal distribution, reflecting the tendency for men and women
to differ in their typical weights (see [p. 11).
10
11
13. What is the relative standing of the mean, median,
and mode in (a) normal, (b) positively skewed, and (c)
negatively skewed distributions?
14.
How can everyday discourse be translated into indices
of central tendency?
15.
What are the advantages of the mean as an index
of central tendency with quantitative variables?
16.
Under what circumstances is the median a better index
of central tendency than the mean with quantitative
variables?
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Summary of the Properties of the Mean, Median, and Mode
The mean (for a sample, M; for a population, x) is:
1.
the balance point of distribution, the point for which
(X - M) = 0;
2.
the preferred measure for relatively symmetrical
distributions and quantitative variables;
3.
the measures with the best sampling stability;
4.
widely used in advanced statistical procedures;
5.
the only measure whose value is dependent on the value of
every score in the distribution;
6.
more sensitive to extreme scores than the median and the
mode and, hence, is not recommended for markedly skewed
distributions or small distributions with an outlier; and
7.
not appropriate for open-ended distributions.
The median is
1.
the point that divides the ordered scores in half;
2.
second to the mean in usefulness;
3.
widely used for markedly skewed distributions or for small
distributions with an outlier because it is sensitive only to
the number rather than to the values of scores above and below
it;
4.
the most stable measure that can be used with open-ended
distributions;
5.
more subject to sampling fluctuation than the mean;
6.
less often used in advanced statistical procedure; and
7.
the most appropriate measure for ordered qualitative
variables.
The mode is
1.
the score (or category) that occurs most often;
2.
the only measure appropriate for unordered qualitative
variables;
3.
the easiest measure to compute;
4.
much more subject to sampling fluctuation that the mean
and the median; and
5.
rarely used in advanced statistical procedures
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