Transcript measures of dispersion

Intro to Statistics Part II
Descriptive Statistics
Ernesto Diaz
Assistant Professor of Mathematics
Copyright © 2016 Brooks/Cole Cengage Learning
14.2
Descriptive Statistics
Copyright © Cengage Learning. All rights reserved.
Descriptive Statistics
Descriptive statistics is concerned with the accumulation
of data, measures of central tendency, and dispersion.
3
Measures of Central Tendency
4
Measures of Central Tendency
When we add up a list of numbers in statistics, we use the
symbol x to mean the sum of all the values that x can
assume.
Similarly, x2 means to square each value that x can
assume, and then add the results; (x)2 means to first add
the values and then square the result.
The symbol  is the Greek capital letter sigma (which is
chosen because S reminds us of “sum”).
The average is the measure that most of us think of when
we hear someone use the word average. It is called the
mean.
5
Measures of Central Tendency
Other statistical measures, called averages or measures
of central tendency, are defined in the following box.
6
Example 3 – Mean, median, and mode for table values
Consider Table 14.5, which shows the number of days one
must wait for a marriage license in the various states in the
United States. What are the mean, the median, and the
mode for these data?
Wait Time for a U.S. Marriage License
Table 14.5
7
Example 3 – Solution
Mean: To find the mean, we could, of course, add all 50
individual numbers, but instead, notice that
0 occurs 25 times, so write
0  25
1 occurs 1 time, so write
11
2 occurs 1 time, so write
21
3 occurs 19 times, so write
3  19
4 occurs 1 time, so write
41
5 occurs 3 times, so write
53
Thus, the mean is
8
Example 3 – Solution
cont’d
Median: Since the median is the middle number and there
are 50 values, the median is the mean of the 25th and 26th
numbers (when they are arranged in order):
25th term is 0
26th term is 1
Mode: The mode is the value that occurs most frequently,
which is 0.
9
Measures of Central Tendency
When finding the mean from a frequency distribution, you
are finding what is called a weighted mean.
10
Example 4 – Find a weighted mean
A sociology class is studying family structures and the
professor asks each student to state the number of children
in his or her family.
The results are summarized in Table 14.6. What is the
average number of children in the families of students in
this sociology class?
Family Data
Table 14.6
11
Example 4 – Solution
We need to find the weighted mean, where x represents
the number of students and w the population (number of
families).
= 2.12
There is an average of two children per family.
12
Measures of Position
13
Measures of Position
The median divides the data into two equal parts, with half
the values above the median and half below the median, so
the median is called a measure of position.
Sometimes we use benchmark positions that divide the
data into more than two parts. Quartiles, denoted by
Q1(first quartile), Q2(second quartile), and Q3(third quartile),
divide the data into four equal parts.
Deciles are nine values that divide the data into ten equal
parts, and percentiles are 99 values that divide the data
into 100 equal parts.
14
Example 5 – Divide exam scores into quartiles
The test results for Professor Hunter’s midterm exam are
summarized in Table 14.7.
Grade Distribution
Table 14.7
Divide these scores into quartiles.
15
Example 5 – Solution
The quartiles are the three scores that divide the data into
four parts. The first quartile is the data value that separates
the lowest 25% of the scores from the remaining scores;
the 2nd quartile is the value that separates the lower 50%
of the scores from the remainder.
Note that the 2nd quartile is the same as the median since
the median divides the scores so that 50% are above and
50% are below. The 3rd quartile is the value that separates
the lower 75% of the scores from the upper 25%.
Begin by noting the number of scores: 4 + 7 + 16 + 3 = 30.
16
Example 5 – Solution
cont’d
First quartile: 0.25(30) = 7.5, so Q1(the first quartile) is the
8th lowest score. From Table 14.7, we see that this score is
69.
Second quartile: Q2 the second quartile score, is the
median, which is the mean of the 15th and 16th scores
from the bottom.
17
Example 5 – Solution
cont’d
Third quartile: 0.75(30) = 22.5, so Q3 (the third quartile
score) is 23 scores from the bottom (or the 8th from the
top). From Table 14.7, we see this score is 85.
Grade Distribution
Table 14.7
18
Measures of Dispersion
19
Measures of Dispersion
The measures we’ve been discussing can help us interpret
information, but they do not give the entire story. For
example, consider these sets of data:
Set A: {8, 9, 9, 9, 10}
Mean:
Median: 9
Mode: 9
Set B: {2, 9, 9, 12, 13}
Mean:
Median: 9
Mode: 9
20
Measures of Dispersion
Notice that, for sets A and B, the measures of central
tendency do not distinguish the data. However, if you look
at the data placed on planks, as shown in Figure 14.29,
you will see that the data in Set B are relatively widely
dispersed along the plank, whereas the data in Set A are
clumped around the mean.
a. A = {8, 9, 9, 9, 10}
b. B = {2, 9, 9, 12, 13}
Visualization of dispersion of sets of data
Figure 14.29
21
Measures of Dispersion
We’ll consider three measures of dispersion: the range,
the standard deviation, and the variance.
22
Example 6 – Find the range
Find the ranges for the data sets in Figure 14.29:
a. Set A = {8, 9, 9, 9,10} b. Set B = {2, 9, 9, 12, 13}
Solution:
Notice from Figure 14.29 that the mean for each of these
sets of data is the same.
a. A = {8, 9, 9, 9, 10}
b. B = {2, 9, 9, 12, 13}
Visualization of dispersion of sets of data
Figure 14.29
23
Example 6 – Solution
cont’d
The range is found by comparing the difference between
the largest and smallest values in the set.
a. 10 – 8 = 2
b. 13 – 2 = 11
24
Measures of Dispersion
The range is used, along with quartiles, to construct a
statistical tool called a box plot.
For a given set of data, a box plot consists of a rectangular
box positioned above a numerical scale, drawn from Q1
(the first quartile) to Q3 (the third quartile).
The median (Q2, or second quartile) is shown as a dashed
line, and a segment is extended to the left to show the
distance to the minimum value; another segment is
extended to the right for the maximum value.
25
Measures of Dispersion
Figure 14.30 shows a box plot for the data in Example 5.
Box plot for grade distribution
Figure 14.30
26
Measures of Dispersion
Sometimes a box plot is called a box-and-whisker plot. Its
usefulness should be clear when you look at Figure 14.31.
box plot shows:
Box plot
Figure 14.31
1. the median (a measure of central tendency);
2. the location of the middle half of the data (represented
by the extent of the box);
27
Measures of Dispersion
3. the range (a measure of dispersion);
4. the skewness (the nonsymmetry of both the box and the
whiskers).
The variance and standard deviation are measures that
use all the numbers in the data set to give information
about the dispersion.
When finding the variance, we must make a distinction
between the variance of the entire population and the
variance of a random sample from the population.
28
Measures of Dispersion
When the variance is based on a set of sample scores, it is
denoted by s2; and when it is based on all scores in a
population, it is denoted by  2 ( is the lowercase Greek
letter sigma).
The variance for a random sample is found by
29
Measures of Dispersion
To understand this formula for the sample variance, we will
consider an example before summarizing a procedure.
Again, let’s use the data sets we worked with in Example 6.
Set A = {8, 9, 9, 9, 10}
Mean is 9.
Set B = {2, 9, 9, 12, 13}
Mean is 9.
30
Measures of Dispersion
Find the deviations by subtracting the mean from each
term:
8 – 9 = –1
2 – 9 = –7
9–9=0
9–9=0
9–9=0
9–9=0
9–9=0
12 – 9 = 3
10 – 9 = 1
13 – 9 = 4
Mean
Mean
If we sum these deviations (to obtain a measure of the total
deviation), in each case we obtain 0, because the positive
and negative differences “cancel each other out.”
31
Measures of Dispersion
Next we calculate the square of each of these deviations:
Set A = {8, 9, 9, 9, 10}
Set B = {2, 9, 9, 12, 13}
(8 – 9)2 = (–1)2 = 1
(2 – 9)2 = (–7)2 = 49
(9 – 9)2 = 02 = 0
(9 – 9)2 = 02 = 0
(9 – 9)2 = 02 = 0
(9 – 9)2 = 02 = 0
(9 – 9)2 = 02 = 0
(12 – 9)2 = 32 = 9
(10 – 9)2 = 12 = 1
(13 – 9)2 = 42 = 16
32
Measures of Dispersion
Finally, we find the sum of these squares and divide by one
less than the number of items to obtain the variance:
Set A:
Set B:
The larger the variance, the more dispersion there is in the
original data.
33
Measures of Dispersion
34
Example 8 – Find the standard deviation for a math test
Suppose that Hannah received the following test scores in
a math class: 92, 85, 65, 89, 96, and 71. Find s, the
standard deviation, for her test scores.
Solution:
Step 1
This is the mean.
35
Example 8 – Solution
Steps 2–4 We summarize these steps in table format:
Score
Square of the Deviation from the Mean
92
(92 – 83)2 = 92 = 81
85
(85 – 83)2 = 22 = 4
65
(65 – 83)2 = (–18)2 = 324
89
(89 – 83)2 = 62 = 36
96
(96 – 83)2 = 132 = 169
71
(71 – 83)2 = (–12)2 = 144
36
Example 8 – Solution
cont’d
Step 5 Divide the sum by 5 (one less than the number of
scores):
We note that this number, 151.6, is called the variance. If
you do not have access to a calculator, you can use the
variance as a measure of dispersion.
However, we assume you have a calculator and can find
the standard deviation.
37
Example 8 – Solution
cont’d
Step 6
38
Interpreting Measures of Dispersion
A main use of dispersion is to compare the
amounts of spread in two (or more) data sets.
A common technique in inferential statistics
is to draw comparisons between populations
by analyzing samples that come from those
populations.
39
39
Example: Interpreting Measures
Two companies, A and B, sell small packs of sugar for
coffee. The mean and standard deviation for samples
from each company are given below. Which company
consistently provides more sugar in their packs?
Which company fills its packs more consistently?
Company A
Company B
xA  1.013 tsp
xB  1.007 tsp
s A  .0021
sB  .0018
40
40
Example: Interpreting Measures
Solution
We infer that Company A most likely provides
more sugar than Company B (greater mean).
We also infer that Company B is more consistent
than Company A (smaller standard deviation).
41
41
Symmetry in Data Sets
The most useful way to analyze a data set often
depends on whether the distribution is
symmetric or non-symmetric. In a
“symmetric” distribution, as we move out from
a central point, the pattern of frequencies is the
same (or nearly so) to the left and right. In a
“non-symmetric” distribution, the patterns to
the left and right are different.
42
© 2008 Pearson Addison-Wesley. All rights reserved
42
Some Symmetric Distributions
43
© 2008 Pearson Addison-Wesley. All rights reserved
43
Non-symmetric Distributions
A non-symmetric distribution with a tail
extending out to the left, shaped like a J, is
called skewed to the left. If the tail
extends out to the right, the distribution is
skewed to the right.
44
© 2008 Pearson Addison-Wesley. All rights reserved
44
Some Non-symmetric Distributions
45
© 2008 Pearson Addison-Wesley. All rights reserved
45
Chebyshev’s Theorem
For any set of numbers, regardless of how
they are distributed, the fraction of them that
lie within k standard deviations of their mean
(where k > 1) is at least
1
1 2
k .
© 2008 Pearson Addison-Wesley. All rights reserved
46
Example: Chebyshev’s Theorem
What is the minimum percentage of the items in a
data set which lie within 3 standard deviations of the
mean?
Solution
With k = 3, we calculate
1
1 8
1  2  1    .889  88.9%.
9 9
3
© 2008 Pearson Addison-Wesley. All rights reserved
47