Download Descriptive Statistics

Descriptive Statistics
Descriptive Statistics can be best viewed as a bag of tricks that allows one to present the essential
information contained in a data set in a way the information can be readily interpreted by the reader.
Included in the bag of tricks will be a number of pictures/graphs - histograms, stem plots, time plots, box
plots - that give us a general picture of the data. There are also a number of formulas/indices - mean,
median, quartiles, standard deviation - that give us a number to describe some facet of the data.
To see what we are talking about with descriptive statistics, let's talk about grades, something that should
be able to catch your interest. [For those who are new to statistics, you might also want to check out
another simpler example based on rolling dice]. You remember after each exam when you wanted to know
the results and what they would mean for your grade in the course. Now let's look at grades from the
instructor's side. Below you will find the complete data set for the first test in a recent course (ECN202).
There were 52 students that received the following scores on their exams. The information for this example
can be found on Histogram and Desc stats tabs of the Descriptive Statistics example.
Grades on Exam 1 in ECN202
Student Grade
Student Grade
Student Grade
Student Grade
1
91.983
14
97.531
27
96.8659
40
89.0001
2
91.7597
15
98.2979
28
76.2859
41
76.5876
3
87.9158
16
70.4242
29
99.6804
42
93.3512
4
77.0586
17
72.6251
30
87.6299
43
88.82
5
98.7479
18
86.9584
31
89.6395
44
77.4919
6
79.8029
19
95.2241
32
85.6969
45
94.7336
7
80.5968
20
91.9544
33
95.2098
46
75.4139
8
77.8953
21
80.2882
34
71.9719
47
86.5368
9
96.1051
22
77.2291
35
92.3448
48
93.7865
10
74.1581
23
93.1482
36
74.4269
49
73.8672
11
83.152
24
75.1727
37
82.7137
50
75.7028
12
91.7678
25
87.5282
38
77.8714
51
73.415
13
78.1368
26
80.501
39
71.828
52
74.7755
While this entire data set may be useful to you for some purposes, I suspect you were not interested in
studying the entire class results. Furthermore, I suspect if you looked at the table for 30 seconds and were
asked to describe what you saw, you would have considerable difficulty capturing the essential features of
the data set. It is for this reason that we have descriptive statistics that allow us to summarize these data
with a few pictures and numbers. As we work our way through these summarizing techniques, you should
realize that in general we will not be able to reproduce the exact data from the summary statistics. Some
information is lost in the process, but the loss should be more than offset by the gains that we have in terms
of insight into the underlying data.
As a starter, the data can be transformed from a table to a graph. One approach would be to simply create a
bar graph as we did in a previous section. If you were to do this by hand the first step in creating the
diagram would be to round off the grades to the nearest whole number and then sort them by score.
1
Test Scores
Score # of tests Score # of tests
70
1
86
1
71
87
2
72
2
88
3
73
2
89
2
74
3
90
1
75
3
91
76
2
92
5
77
4
93
2
78
3
94
1
79
95
3
80
2
96
1
81
2
97
1
82
98
2
83
2
99
1
84
100
1
If we take these data and use Excel to create a column graph with score on the horizontal and # of tests on
the vertical and graph how many students received each score, we get the following diagram. In this
example we can see that four students received a 77 and 5 students received a 92. We could also create a
second graph that will look exactly like the first except we will graph the relative frequency against the
scores. For example, a score of 92 is received by approximately 10 percent of the students (5 of 52). These
two graphs give us a picture of the distribution of ECN202 grades for the entire class.
You could also use the data analysis tools in excel to create a Histogram that looks very similar to what you
created above. The most important difference is you do not need to sort the data to construct the graph. To
see how the Histogram is constructed you should examine the Histogram tab of the Descriptive Statistics
example. The first step was to create a column of grades and then select Data Analysis located in Tools.
Once the table has been created in Excel you select Data Analysis from the Tool menu and then select
Histogram. The dialogue box lets you know that you need to tell it where the data are, where the categories
are (Bins), and where you want the results. You can identify the input by simply dragging the cursor down
the column of scores and then you tell it where the Bin data are by dragging the cursor down the column of
numbers. Finally you give the location for the output by identifying the cell that will be the top left corner
of the data output. The results appear below and on the Histogram tab beginning in cell D3. [Note that I did
a bit of double clicking on things to clean it up a bit. You notice the difference between the two Histograms
on this page].
2
Bin
73.0
76.0
79.0
82.0
85.0
88.0
91.0
94.0
97.0
Here the grades are grouped. We have four students who received below 73 and six students who received
between 85 and 88.
Before you leave histograms behind, you should check out an on-line interactive example of a histogram,
and David Lane's site. Once you are comfortable with looking at the data set graphically, we can then look
at specific features of the distribution of scores. Generally there are four features of the distribution we are
concerned with: modality, symmetry, central tendency, and variability. Of the four, modality and symmetry
are the easiest to visualize. Any value at which the frequency curve or relative frequency curve reaches a
peak is called a mode. Most distributions in practice have one peak and are described as "unimodal." A
distribution with two peaks is called "bimodal." The distribution of grades pictured above has a number of
modes and would not be easily characterized by the mode.
A distribution is said to be symmetric if the relative frequency is the same distance either side of its center.
In the above example, if the midpoint of the distribution were 85, then symmetry would imply that the
same number of students received a grade 90 as a grade of 80, and the number that received a grade of 75
equaled the number that received a 95. The mean and median, concepts that we will discuss in the next
section, are equal in a symmetric distribution. An asymmetric frequency distribution is skewed to the left if
the lower tail is longer than the upper tail and skewed to the right if the upper tail is longer than the lower
tail. To understand the concept of symmetry fully, however, we need to look at the concepts of Central
Tendency and Variability.
As you work your way through the statistical analysis, keep in mind that one of the wonders of modern
computers and statistical software is they can make statistical analysis quite simple. Consider the situation
where you have been asked to analyze Voter Turnout data in the Presidential Elections as provided by the
Federal Elections Commission (FEC) appears below. With these data tools you can then highlight the data
in the second column and select Data Analysis in the Tools menu of Excel. This analysis can be found on
the Desc stats tab of the Descriptive Statistics example. On that worksheet you will also find the results of
running descriptive statistics on the grade data.
3
1992
Voter Turnout (%)
STATE
Alabama
55.2%
Alaska
65.4%
Arizona
54.1%
Arkansas
53.8%
California
49.1%
Colorado
62.7%
Connecticut
63.8%
Delaware
55.2%
District of Columbia
49.6%
…..
…..
Utah
65.2%
Vermont
67.5%
Virginia
52.8%
Washington
59.9%
West Virginia
50.7%
Wisconsin
69.0%
Wyoming
62.3%
UNITED STATES
55.2%
Once you have selected the Data Analysis a dialogue box will appear and you should select Descriptive
Statistics. You will then get a new dialogue box and in the Input Range you should type in the first and last
cell in the third column separated by a colon - in my example it was I4:I54. You will also want to tell it
where you want the output. In this example it went on the same spreadsheet with the top left cell being K4
that is what you put in the dialogue box. Included here is the output you will get that includes the basic
measures of central tendency (mean, median, mode) and variability (standard deviation, variance, and
range). You also note the minimum and maximum values and the number of observations (count). As for
measures of central tendency, the mean was 58% and the median was 59 %. The voter participation rate
ranged from approximately 42% to 72% with a standard deviation 7.3%.
Column1
Mean
0.581318
Standard Error
0.010279
Median
Mode
0.5894
0.6515
Standard Deviation
0.073408
Sample Variance
0.005389
Kurtosis
-0.84522
Skewness
-0.03072
Range
0.3004
Minimum
0.4194
Maximum
0.7198
Sum
Count
29.6472
51
4
A second example, one based on the earlier grade data for ECN202, can also be found on the Desc stats tab
of the Descriptive Statistics example. The Desc stats sheet provides the summary statistics for a set of 52
grades that were generated by selecting Data Analysis under the Tools menu and then choosing Descriptive
Statistics. The input is specified by simply highlighting the column of data and the output is specified to
appear in cell D4.
To understand the concepts that appear in this table you should look at the concepts of central tendency and
variability.
Central Tendency
What's the average? We have heard the question many times: travelers will ask about on-time averages,
university administrators will ask about retention rates (average number of students that return after their
first year), investors will ask about average rates of return, sports fans may ask about batting averages,...
But how do we go about answering the questions? How do we compute the averages? The first thing to
realize is that when people are talking about the average they are talking about a measure of central
tendency. In this section we will talk about three measures of central tendency, the mean, median, and
mode. To better understand the difference between the three measures, let's return to our example of
grades. For an on-line discussion of measures of central tendency you should check out the UCLA On-line
Statistics Course, DAU the Stat refresher, and Hyperstat Online by David Lane at Rice University.
To demonstrate the measures of central tendency we will use our two grading examples. The frequency
distribution of the exam scores for ECN202 is repeated in the first diagram below and the histogram for the
grades in ECN201 follows it.
Mean: This is what people generally mean when they say average. The mean is the arithmetic mean of all
the data and is defined as the sum of all possible values divided by the number of observations. In the table
below we have the grade data for the class that had been divided into four groups. For each group we
added up the grades to get a total for the grades and then divided the total by 13, the number of students in
each group. If we then add the four group averages and then divide this sum by the number of groups, you
get the average of 84.454. This is the same number you would have obtained if you had simply added up all
52 grades and divided by 52 [Note: this procedure works here only because the groups all had the same
number of students. If the groups were of different size, then you would have needed to weight the groups
by their relative size.]
In terms of the frequency distribution above, you see there are not any actual scores of 84 or 85, but those
scores are in the middle of the distribution of grades. If there were more grades in the higher range, then
this would drag up the mean score.
5
Grades on ECN202 Exam 1
Student
Grade
Student
Grade
Student
Grade
Student
Grade
1
91.983
14
97.531
27
96.8659
40
89.0001
2
91.7597
15
98.2979
28
76.2859
41
76.5876
3
87.9158
16
70.4242
29
99.6804
42
93.3512
4
77.0586
17
72.6251
30
87.6299
43
88.82
5
98.7479
18
86.9584
31
89.6395
44
77.4919
6
79.8029
19
95.2241
32
85.6969
45
94.7336
7
80.5968
20
91.9544
33
95.2098
46
75.4139
8
77.8953
21
80.2882
34
71.9719
47
86.5368
9
96.1051
22
77.2291
35
92.3448
48
93.7865
10
74.1581
23
93.1482
36
74.4269
49
73.8672
11
83.152
24
75.1727
37
82.7137
50
75.7028
12
91.7678
25
87.5282
38
77.8714
51
73.415
26
80.501
39
71.828
52
74.7755
13
78.1368
Total
1109.08
1106.88
1102.16
1073.48
Group
Average
85.3138
85.1448
84.7819
82.5756
Average
84.454
One notable feature is the difference between the class average and the average for the groups. As we will
see in the inferential statistics section, group averages can be viewed as averages based on a sample of the
entire student population, and differences persist.
Median: The median is the midpoint in the distribution of grades. At the median there are as many people
above as there are below the median score. In this ECN202 class, the median grade was 84.42, slightly
below the mean value.
Grades on ECN202 Exam 1
Grade
Student
70.4
1
76.6
14
85.7
27
92.3
40
71.8
2
77.1
15
86.5
28
93.1
41
72.0
3
77.2
16
87.0
29
93.4
42
72.6
4
77.5
17
87.5
30
93.8
43
73.4
5
77.9
18
87.6
31
94.7
44
73.9
6
77.9
19
87.9
32
95.2
45
74.2
7
78.1
20
88.8
33
95.2
46
74.4
8
79.8
21
89.0
34
96.1
47
74.8
9
80.3
22
89.6
35
96.9
48
75.2
10
80.5
23
91.8
36
97.5
49
75.4
11
80.6
24
91.8
37
98.3
50
75.7
12
82.7
25
92.0
38
98.7
51
76.3
13
83.2
26
92.0
39
99.7
52
Median = (83.2+85.7)
Grade Student Grade Student Grade Student
84.4244
6
The median for the population of ECN201 grades was 22 while the median for the sample was 21.5.
Mode: The mode refers to the most frequently observed number. If we look at the scores and round them
to whole numbers and sort them we can find the number 77 appears four times and the number 92 appears
5 times. These would be the modes in the score distribution.
Grades Grades Grades Grades
70
77
86
92
72
77
87
93
72
77
87
93
73
77
88
94
73
78
88
95
74
78
88
95
74
78
89
95
74
80
89
96
75
80
90
97
75
81
92
98
75
81
92
98
76
83
92
99
76
83
92
100
In this example we have looked at three measures of central tendency, each of which gives us a bit of
information on the class' performance on the exam. The fact that they are all different provides the
sophisticated observer with additional information. For example, if the mean, median, and mode are the
same, you are most likely looking at a symmetric distribution. If the mean is above the median, then you
probably have an outlier to the right, which would give you a long upper tail that would translate into a
skew to the right in the distribution. This is what we would be likely to see in income statistics if there were
a few very wealthy people in the sample. If the mean tends to be below the median, then there is a long
lower tail - what you would expect to see in a distribution of grades when there was someone with a very
low grade. Now it is time to move on to a discussion of variability.
Variability
As important as measures of central tendency are, they do not provide us with a complete picture of the
underlying scores. We may know what the averages are, but what about the distribution of grades - the
'spread' of the data? Will you feel the same about two possible grading schemes with the same average one where there will be only five A's and five F's and a second where there will no A's or F's? Experience
suggests most students would care, although there is no agreement as to the preferred scheme. Those with
good academic records tend to favor the first distribution while those with weak records tend to favor the
second.
The differences between the two grade distributions will be captured in the variability of the scores. As
with the previous discussion of central tendency, there are a number of measures of variation. We will look
at range, variance, and standard deviation. For an on-line discussion of measures of variability you should
check out the UCLA On-line Statistics Course, the Electronic Textbook, Introductory Statistics: Concepts,
Models, and Applications Copyright 1996 by David W. Stockburger, DAU the Stat refresher, and
Hyperstat Online by David Lane at Rice University.
To demonstrate the measures of variability we will use our grading example. The frequency diagram of the
exam scores for ECN202 is repeated below.
7
Range: This is the easy one. Once you have your data sorted to identify your median, simply look at the
highest and lowest values. The range is the difference between the lowest and highest values. In the
ECN202 grade example, grades range from 70 to 100. No one received a score higher than 100 or lower
than 70.
Variance: How likely is it that we would get a score close to the average? Did most of the students receive
similar scores or were they spread out roughly equally over the entire range. The variance can be thought
of as a measure of variability derived by a two-step process. In the first we compute a new variable that
equals the test score minus the mean score (approximately 84.5). The result appears in the second column
below. If we add the deviation in the first row (7.9) to the mean (84.5) we get the score (92.3) [Note: there
may be a small difference due to rounding].
ECN202 Grades
Score
92.3
85.7
93.1
76.6
86.5
70.4
77.1
71.8
93.4
87
93.8
77.2
87.5
94.7
72
…
95.2
96.1
73.4
77.9
88.8
Deviation
7.9
1.2
8.7
-7.9
2.1
-14
-7.4
-12.6
8.9
2.5
9.3
-7.2
3.1
10.3
-12.5
…
10.8
11.7
-11
-6.6
4.4
Deviation 2
62.264
1.54463
75.5893
61.8804
4.33774
196.835
54.6925
159.417
79.1603
6.2721
87.0956
52.1999
9.45035
105.67
155.804
…
115.994
135.747
121.861
43.0169
19.0616
Score
79.8
89.6
97.5
98.3
74.2
74.4
80.3
91.8
91.8
98.7
74.8
80.5
80.6
92
99.7
…
…
Deviation
-4.7
5.2
13.1
13.8
-10.3
-10
-4.2
7.3
7.3
14.3
-9.7
-4
-3.9
7.5
15.2
…
76.3
89
96.9
73.9
78.1
-8.2
4.5
12.4
-10.6
-6.3
Mean
84.454
Variance
76.45
Standard deviation
8.74
Deviation 2
21.633
26.8889
171.007
191.654
106.006
100.544
17.3537
53.3722
53.4916
204.316
93.6733
15.6263
14.8779
56.2552
231.843
…
66.7191
20.667
154.054
112.082
39.9078
8
In the second step we square all of the deviation terms in column 2 that generates column 3. We now add
these squared terms and divide by the number of observations to get the variance. All other things equal,
the greater the variance the greater the spread of the scores.
What would the distribution look like if the variance were larger? What you would see would be fewer
observations close to the mean and more observations near the upper and lower limits.
Standard Deviation: There is one problem with the variance - it is influenced by the size of the variable
being analyzed which will make comparisons of different score distributions impossible. For example, if
one teacher used a 4-point scale and another used the 100-point scale, then there would be a real problem
comparing the variability in grades for the two classes. To allow for this comparability, we can 'normalize'
the variance by taking its square root. The result is the standard deviation, which in the ECN202 example is
8.74 points. In the ECN201 example, the standard deviations for the entire class and the sample are 4.37
and 4.11.
In the next section where we examine probability and probability distributions, the standard deviation will
take on special significance.
9