Math 211 Introduction to Statistics
Chapter 4 Measures of Dispersion
Dispersion: The degree to which numerical raw data tend to spread about an average value is called the Dispersion, or Variation of the data. The most common measures of dispersion is the range, mean deviation, semi-interquartile range, and standard deviation.
The Range: The difference between the largest and smallest numbers in the set.
The Mean Deviation: (Average deviation) The mean deviation of a set of numbers is denoted by and defined as
where is the arithmetic mean, is the absolute value of the deviations of from .
If occur with frequencies respectively, the mean deviation can be written as
where . This form is useful for grouped data, where ’s represent class marks and ’s are the corresponding class frequencies.
The Semi-Interquartile Range: (Quartile Deviation)
The 10-90 Percentile Range:
Semi- 10-90 Percentile Range:
The Standard Deviation: The standard deviation of a set of numbers is denoted by and defined as .
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If occur with frequencies respectively, the standard deviation can be written as
where . This form is useful for grouped data, where ’s represent class marks and ’s are the corresponding class frequencies.
The Variance: The variance of a set of numbers is denoted by and defined as .
Properties of the Standard Deviation
(1) The standard deviation can be defined as
. is minimum when .
(2) For moderately skewed distributions, the percentages below may hold approximately.
For normal distributions,
1 s.d. on either 2 s.d. on either side
side of the mean of the mean
3 s.d. on either side
of the mean
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Short methods for computing the standard deviation
(1)
(2) If are the deviations of from some arbitrary constant , then
(3) (Coding Method) When data are grouped into a frequency distribution whose class intervals have equal size c, we have or where , then
Example 1. Determine the percentage of the students with grades that fall within their ranges
(a)
(b) .
Given,
Grades / No.ofstudents / / / / /
10-19 / 2 / 14.5 / -3 / -6 / 9 / 18
20-29 / 5 / 24.5 / -2 / -10 / 4 / 20
30-39 / 8 / 34.5 / -1 / -8 / 1 / 8
40-49 / 11 / 44.5 / 0 / 0 / 0 / 0
50-59 / 8 / 54.5 / 1 / 8 / 1 / 8
60-69 / 5 / 64.5 / 2 / 10 / 4 / 20
70-79 / 2 / 74.5 / 3 / 6 / 9 / 18
N=41
Let A=44.5, and c=10.
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(a)
The number of students in the range is, 8+11+8=27.
The percentage of grades is .
(b)
The number of students in the range is, .
The percentage of grades is .
Example 2. Consider the following frequency distribution to compute using coding method for .
Class boundaries / Freq.() / / / / / / /154.5-158.5 / 2 / 156.5 / -3 / -6 / 9 / 18 / -13.8 / 27.6
158.5-162.5 / 3 / 160.5 / -2 / -6 / 4 / 12 / -9.8 / 29.4
162.5-166.5 / 8 / 164.5 / -1 / -8 / 1 / 8 / -5.8 / 46.4
166.5-170.5 / 16 / 168.5 / 0 / 0 / 0 / 0 / -1.8 / 34.2
170.5-174.5 / 12 / 172.5 / 1 / 12 / 1 / 12 / 2.2 / 26.4
174.5-178.5 / 9 / 176.5 / 2 / 18 / 4 / 36 / 6.2 / 55.8
178.5-182.5 / 5 / 180.5 / 3 / 15 / 9 / 45 / 10.2 / 51
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Empirical Relation between Measures of Dispersions
For moderately skewed distributions, we have the empirical formulae
.
Absolute and Relative Dispersion; coefficient of variation
Absolute dispersion is the actual variation.
Relative Dispersion.
If absolute Dispersion and average, then
.
Example 3. On a final examination in Statistics, the mean grade of a group of 150 students was 78 and the standard deviation was 8.0. In Calculus, however, the mean grade of the group was 73 and the standard deviation was 7.6. Which subject has the greater
(a) absolute dispersion
(b) relative dispersion
(a) The absolute dispersion of Statistics is and of Calculus .
Therefore, the subject Calculus has smaller absolute dispersion.
(b) Coefficients of variation are
Standardized Variable: Standard Scores
The variable that measures the deviation from the mean in units of the standard deviation is called a standardized variable and is given by .
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Example 4. A student received a grade of 84 on a final examination in Mathematics for which the mean grade was 76 and the standard deviation was 10. On the final examination in Physics, for which the mean grade was 82 and the standard deviation was16, she received a grade of 90. In which subject was her relative standing higher?
, ,
Since ,
Therefore the relative standing of the student is higher in Mathematics.
Example 5. Find the mean deviation of the numbers 2,2,4,6,7,8,9,12.
First we need to find . That is,
Then the mean deviation is,
Example 6. Find (a) the Semi-Interquartile Range
(b) 10-90 Percentile Range for the data given in Example 2.
(a) The semi-interquartile range is .
Hence 50% of the cases lie between 166.69 and 174.75. So the measure of tendency is . In other words, 50% of the cases lie in the range 170.724.03.
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(b) The 10-90 Percentile Range
.
and
We conclude that 80% of the cases lie in the range 170.9558.705.
Example 7. Find the standard deviation and the variance of the following set of numbers:
6, 8, 12, 7, 4, 5, 5, 10, 9, 8
6 / -1.4 / 1.968 / 0.6 / 0.36
12 / 4.6 / 21.16
7 / -0.4 / 0.16
4 / -3.4 / 11.56
5 / -2.4 / 5.76
5 / -2.4 / 5.76
10 / 2.6 / 6.76
9 / 1.6 / 2.56
8 / 0.6 / 0.36
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Example 8: Consider the following frequency distribution.
classes / frequency / / / / /10-14 / 7 / 12 / -2 / 4 / -14 / 28
15-19 / 11 / 17 / -1 / 1 / -11 / 11
20-24 / 14 / 22 / 0 / 0 / 0 / 0
25-29 / 13 / 27 / 1 / 1 / 13 / 13
30-34 / 5 / 32 / 2 / 4 / 10 / 20
Total 50 / /
Use the Coding Method to compute and .
The mean value is
The standard deviation is .
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