12.3 – Measures of Dispersion Dispersion is another analytical method to study data. Two of the most common measures of dispersion are the range and the standard deviation. Range For any set of data, the range of the set is given by the following formula: Range = (greatest value in set) – (least value in set). A common technique in inferential statistics is to draw comparisons between populations by analyzing samples that come from those populations. A main use of dispersion is to compare the amounts of spread in two (or more) data sets.
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12.3 – Measures of DispersionDispersion is another analytical method to study data.
Two of the most common measures of dispersion are the range and the standard deviation.
RangeFor any set of data, the range of the set is given by the following formula:
Range = (greatest value in set) – (least value in set).
A common technique in inferential statistics is to draw comparisons between populations by analyzing samples that come from those populations.
A main use of dispersion is to compare the amounts of spread in two (or more) data sets.
Example:
Set A 1 2 7 12 13
Set B 5 6 7 8 9
The two sets below have the same mean and median (7). Find the range of each set.
Range of Set A:
12.3 – Measures of DispersionRange
13 – 1 = 12
Range of Set A: 9 – 5 = 4
One of the most useful measures of dispersion is the standard deviation.
12.3 – Measures of DispersionStandard Deviation
It is based on deviations from the mean of the data.
Find the deviations from the mean for all data values of the sample 1, 2, 8, 11, 13.The mean is 7. To find each deviation, subtract the mean from each data value.
Data Value 1 2 8 11 13Deviation – 6 – 5 1 4 6
The sum of the deviations is always equal to zero.
The sample standard deviation is found by calculating the square root of the variance.
12.3 – Measures of DispersionStandard Deviation
Calculating the Sample Standard Deviation
2( ).
1x x
sn
The sample standard deviation is denoted by the letter s.
The variance is found by summing the squares of the deviations and dividing that sum by n – 1 (since it is a sample instead of a population).
The standard deviation of a population is denoted by .
6. Take the square root of the quotient in Step 5.
12.3 – Measures of DispersionStandard Deviation
Calculating the Sample Standard Deviation
1. Calculate the mean of the numbers.2. Find the deviations from the mean.3. Square each deviation.4. Sum the squared deviations.5. Divide the sum in Step 4 by n – 1.
Find the standard deviation of the sample set {1, 2, 8, 11, 13}.
Data Value 1 2 8 11 13Deviation
Sum of the (Deviations)2 =
12.3 – Measures of DispersionStandard Deviation
Calculating the Sample Standard DeviationExample:
= 7
– 6 – 5 1 4 6(Deviation)2 16 1 25 36 36
36 + 25 + 1 + 16 + 36 = 114
Sum of the (Deviations)2 =
12.3 – Measures of DispersionStandard Deviation
Calculating the Sample Standard Deviation
36 + 25 + 1 + 16 + 36 = 114Divide 114 by n – 1 with n = 5:
1145 – 1
= 28.5
Take the square root of 28.5: 5.34
The sample standard deviation of the data is 5.34.
Example: Interpreting MeasuresTwo 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
1.013 tspAx 1.007 tspBx
.0021As .0018Bs
12.3 – Measures of DispersionStandard Deviation
Example: Interpreting Measures
The sample mean for Company A is greater than the sample mean of Company B.
Company A Company B
1.013 tspAx 1.007 tspBx
.0021As .0018Bs
12.3 – Measures of DispersionStandard Deviation
Which company consistently provides more sugar in their packs?
The inference can be made that Company A provides more sugar in their packs.
Example: Interpreting Measures
Which company fills its packs more consistently?
Company A Company B
1.013 tspAx 1.007 tspBx
.0021As .0018Bs
12.3 – Measures of DispersionStandard Deviation
The standard deviation for Company B is less than the standard deviation for Company A.The inference can be made that Company B fills their packs more closer to their mean than Company A.
Chebyshev’s TheoremFor 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
211
.k
12.3 – Measures of Dispersion
What is the minimum percentage of the items in a data set which lie within 2, and 3 standard deviations of the mean?
75% 88.9%
Coefficient of VariationThe coefficient of variation expresses the standard deviation as a percentage of the mean.
12.3 – Measures of Dispersion
It is not strictly a measure of dispersion as it combines central tendency and dispersion.
For any set of data, the coefficient of variation is given by
for a sample or
for a population.
100sVx
100V
Example: Comparing Samples
Compare the dispersions in the two samples A and B. A: 12, 13, 16, 18, 18, 20B: 125, 131, 144, 158, 168, 193
Coefficient of Variation12.3 – Measures of Dispersion
Sample A Sample B16.167Ax 153.167Bx 3.125As 25.294Bs 19.3AV 16.5BV
Sample B has a larger dispersion than sample A, but sample A has the larger relative dispersion (coefficient of variation).
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