1. Measures of Dispersion
Chapter 3 Section 2
Measures of
Dispersion

2. Chapter 3 – Section 2 Comparing two sets of data
The measures of central tendency (mean, median, mode) measure the differences between the “average” or “typical” values between two sets of data
The measures of dispersion in this section measure the differences between how far “spread out” the data values are
Comparing two sets of data
Comparing two sets of data
The measures of central tendency (mean, median, mode) measure the differences between the “average” or “typical” values between two sets of data

3. Chapter 3 – Section 2
The range of a variable is the largest data value minus the smallest data value
Compute the range of
6, 1, 2, 6, 11, 7, 3, 3
The largest value is 11
The smallest value is 1
Subtracting the two … 11 – 1 = 10 … the range is 10
The range of a variable is the largest data value minus the smallest data value
Compute the range of
6, 1, 2, 6, 11, 7, 3, 3
The largest value is 11
The smallest value is 1
The range of a variable is the largest data value minus the smallest data value
Compute the range of
6, 1, 2, 6, 11, 7, 3, 3
The range of a variable is the largest data value minus the smallest data value

4. Chapter 3 – Section 2
The range only uses two values in the data set – the largest value and the smallest value
The range is not resistant
If we made a mistake and
6, 1, 2
was recorded as
6000, 1, 2
The range is now ( 6000 – 1 ) = 5999
The range only uses two values in the data set – the largest value and the smallest value
The range is not resistant
The range only uses two values in the data set – the largest value and the smallest value
The range is not resistant
If we made a mistake and
6, 1, 2
was recorded as
6000, 1, 2

5. Chapter 3 – Section 2
The variance is based on the deviation from the mean
( xi – μ ) for populations
( xi – ) for samples
To treat positive differences and negative differences, we square the deviations
( xi – μ )2 for populations
( xi – )2 for samples
The variance is based on the deviation from the mean
( xi – μ ) for populations
( xi – ) for samples

6. Chapter 3 – Section 2
The population variance of a variable is the sum of these squared deviations divided by the number in the population
The population variance is represented by σ2
Note: For accuracy, use as many decimal places as allowed by your calculator
The population variance of a variable is the sum of these squared deviations divided by the number in the population
The population variance of a variable is the sum of these squared deviations divided by the number in the population

7. Chapter 3 – Section 2 Compute the population variance of 6, 1, 2, 11
Compute the population mean first
μ = ( ) / 4 = 5
Now compute the squared deviations
(1–5)2 = 16, (2–5)2 = 9, (6–5)2 = 1, (11–5)2 = 36
Average the squared deviations
( ) / 4 = 15.5
The population variance σ2 is 15.5
Compute the population variance of
6, 1, 2, 11
Compute the population mean first
μ = ( ) / 4 = 5
Now compute the squared deviations
(1–5)2 = 16, (2–5)2 = 9, (6–5)2 = 1, (11–5)2 = 36
Compute the population variance of
6, 1, 2, 11
Compute the population mean first
μ = ( ) / 4 = 5
Compute the population variance of
6, 1, 2, 11

8. Chapter 3 – Section 2
The sample variance of a variable is the sum of these squared deviations divided by one less than the number in the sample
The sample variance is represented by s2
We say that this statistic has n – 1 degrees of freedom
The sample variance of a variable is the sum of these squared deviations divided by one less than the number in the sample

9. Chapter 3 – Section 2 Compute the sample variance of 6, 1, 2, 11
Compute the sample mean first
= ( ) / 4 = 5
Now compute the squared deviations
(1–5)2 = 16, (2–5)2 = 9, (6–5)2 = 1, (11–5)2 = 36
Average the squared deviations
( ) / 3 = 20.7
The sample variance s2 is 20.7
Compute the sample variance of
6, 1, 2, 11
Compute the sample mean first
= ( ) / 4 = 5
Now compute the squared deviations
(1–5)2 = 16, (2–5)2 = 9, (6–5)2 = 1, (11–5)2 = 36
Compute the sample variance of
6, 1, 2, 11
Compute the sample mean first
= ( ) / 4 = 5
Compute the sample variance of
6, 1, 2, 11

10. Chapter 3 – Section 2
Why are the population variance (15.5) and the sample variance (20.7) different for the same set of numbers?
In the first case, { 6, 1, 2, 11 } was the entire population (divide by N)
In the second case, { 6, 1, 2, 11 } was just a sample from the population (divide by n – 1)
These are two different situations
Why are the population variance (15.5) and the sample variance (20.7) different for the same set of numbers?
In the first case, { 6, 1, 2, 11 } was the entire population (divide by N)
In the second case, { 6, 1, 2, 11 } was just a sample from the population (divide by n – 1)
Why are the population variance (15.5) and the sample variance (20.7) different for the same set of numbers?
In the first case, { 6, 1, 2, 11 } was the entire population (divide by N)
Why are the population variance (15.5) and the sample variance (20.7) different for the same set of numbers?

11. Chapter 3 – Section 2 Why do we use different formulas?
The reason is that using the sample mean is not quite as accurate as using the population mean
If we used “n” in the denominator for the sample variance calculation, we would get a “biased” result
Bias here means that we would tend to underestimate the true variance

12. Chapter 3 – Section 2
The standard deviation is the square root of the variance
The population standard deviation
Is the square root of the population variance (σ2)
Is represented by σ
The sample standard deviation
Is the square root of the sample variance (s2)
Is represented by s
The standard deviation is the square root of the variance
The standard deviation is the square root of the variance
The population standard deviation
Is the square root of the population variance (σ2)
Is represented by σ

13. Chapter 3 – Section 2 If the population is { 6, 1, 2, 11 }
The population variance σ2 = 15.5
The population standard deviation σ =
If the sample is { 6, 1, 2, 11 }
The sample variance s2 = 20.7
The sample standard deviation s =
The population standard deviation and the sample standard deviation apply in different situations
If the population is { 6, 1, 2, 11 }
The population variance σ2 = 15.5
The population standard deviation σ =
If the population is { 6, 1, 2, 11 }
The population variance σ2 = 15.5
The population standard deviation σ =
If the sample is { 6, 1, 2, 11 }
The sample variance s2 = 20.7
The sample standard deviation s =

14. Chapter 3 – Section 2
The standard deviation is very useful for estimating probabilities

15. Chapter 3 – Section 2 The empirical rule
If the distribution is roughly bell shaped, then
Approximately 68% of the data will lie within 1 standard deviation of the mean
Approximately 95% of the data will lie within 2 standard deviations of the mean
Approximately 99.7% of the data (i.e. almost all) will lie within 3 standard deviations of the mean
The empirical rule
If the distribution is roughly bell shaped, then
Approximately 68% of the data will lie within 1 standard deviation of the mean
Approximately 95% of the data will lie within 2 standard deviations of the mean
The empirical rule
If the distribution is roughly bell shaped, then
Approximately 68% of the data will lie within 1 standard deviation of the mean
The empirical rule
If the distribution is roughly bell shaped, then

16. Chapter 3 – Section 2
For a variable with mean 17 and standard deviation 3.4
Approximately 68% of the values will lie between (17 – 3.4) and ( ), i.e and 20.4
Approximately 95% of the values will lie between (17 – 2  3.4) and (  3.4), i.e and 23.8
Approximately 99.7% of the values will lie between (17 – 3  3.4) and (  3.4), i.e. 6.8 and 27.2
A value of 2.1 and a value of 33.2 would both be very unusual
For a variable with mean 17 and standard deviation 3.4
Approximately 68% of the values will lie between (17 – 3.4) and ( ), i.e and 20.4
Approximately 95% of the values will lie between (17 – 2  3.4) and (  3.4), i.e and 23.8
Approximately 99.7% of the values will lie between (17 – 3  3.4) and (  3.4), i.e. 6.8 and 27.2
For a variable with mean 17 and standard deviation 3.4
Approximately 68% of the values will lie between (17 – 3.4) and ( ), i.e and 20.4
For a variable with mean 17 and standard deviation 3.4
For a variable with mean 17 and standard deviation 3.4
Approximately 68% of the values will lie between (17 – 3.4) and ( ), i.e and 20.4
Approximately 95% of the values will lie between (17 – 2  3.4) and (  3.4), i.e and 23.8

17. Chapter 3 – Section 2
Chebyshev’s inequality gives a lower bound on the percentage of observations that lie within k standard deviations of the mean (where k > 1)
This lower bound is
An estimated percentage
The actual percentage for any variable cannot be lower than this number
Therefore the actual percentage must be this value or higher
Chebyshev’s inequality gives a lower bound on the percentage of observations that lie within k standard deviations of the mean (where k > 1)
Chebyshev’s inequality gives a lower bound on the percentage of observations that lie within k standard deviations of the mean (where k > 1)
This lower bound is
An estimated percentage
The actual percentage for any variable cannot be lower than this number

18. Chapter 3 – Section 2 Chebyshev’s inequality
For any data set, at least
of the observations will lie within k standard deviations of the mean, where k is any number greater than 1

19. Chapter 3 – Section 2
How much of the data lies within 1.5 standard deviations of the mean?
From Chebyshev’s inequality
so that at least 55.6% of the data will lie within 1.5 standard deviations of the mean

20. Chapter 3 – Section 2
If the mean is equal to 20 and the standard deviation is equal to 4, how much of the data lies between 14 and 26?
14 to 26 are 1.5 standard deviations from 20
so that at least 55.6% of the data will lie between 14 and 26

21. Summary: Chapter 3 – Section 2
Range
The maximum minus the minimum
Not a resistant measurement
Variance and standard deviation
Measures deviations from the mean
Empirical rule
About 68% of the data is within 1 standard deviation
About 95% of the data is within 2 standard deviations