# 2.4 Measures of Variation - PowerPoint PPT Presentation

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2.4 Measures of Variation. The Range of a data set is simply: Range = (Max. entry) – (Min. entry). Deviation. The deviation of an entry, x , is the difference between the entry and the mean, , of the data set. Mean = Deviation of x = x - . Population Variance.

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2.4 Measures of Variation

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## 2.4 Measures of Variation

The Range of a data set is simply:

Range = (Max. entry) – (Min. entry)

### Deviation

• The deviation of an entry, x, is the difference between the entry and the mean, , of the data set.

• Mean =

• Deviation of x = x - 

### Population Variance

• We are not going to be talking much about the Population Variance. We will be talking more about the Sample Variance.

• Population Variance is found by:

• Find the mean of the population  (note the symbol)

• Find the deviation of each point by subtracting the mean from each data point

• Square the differences

• Add all the squares up

• Divide by the total number of data points in the population

• Population Variance:

### Population Standard Deviation

• The Population Standard Deviation is the square root of the Population Variance.

### Sample Variance

• We will be talking mostly about the Sample Variance.

• Why?

• Sample Variance is found by:

• Find the mean of the sample:

• Find the deviation by subtracting the mean of the sample from each data point

• Square the differences

• Add all the squares up

• Divide by the total number of data points in the sample minus 1.

• Sample Variance:

### Sample Standard Deviation

• The Sample Standard Deviation is the square root of the Sample Variance.

### Example

• Find the standard deviation of the following sample:

### Example

• Find the standard deviation of the following sample:

### Example

• Find the standard deviation of the following sample:

### Example

• Find the standard deviation of the following sample:

### Example

• Find the standard deviation of the following sample:

What will be the sum of this column?

### Example

• Find the standard deviation of the following sample:

What will be the sum of this column?

It will always be zero

### Example

• Find the standard deviation of the following sample:

### Example

• Find the standard deviation of the following sample:

### Standard Deviation

• The TI calculators can calculate both standard deviations quickly:

• Stats

• Calc

• 1-Var Stats

• Enter the list you want to use

• Enter

### Standard Deviation

• This gives:

• The mean of the data:

• The sum of all of the data:

• The sum of the squares of all the data:

• Sample standard deviation:

• Population standard deviation:

• The number of data points:

• The smallest data point value: minX

• Etc.

### Standard Deviation

• What does Standard Deviation represent?

• It is a measure of the distance from the mean.

• It is a measure of how far the data is from the mean.

• It is a measure of the spread of data.

• The larger the Standard Deviation, the more spread out the data is.

### Standard Deviation

• Calculate the mean, range, and standard deviations for 8 units at a value of 7:

• Mean = 7

• Range = 0

• Population and Sample Standard Deviations = 0, why?

• There is no spread in the data. It is all the exact same number

### Standard Deviation

• Calculate the mean, range, and standard deviations for 4 units each at 6 and 8:

• Mean = 7

• Range = 2

• Population Standard deviation = 1, why?

• The data is an average of one unit from the mean

• Sample Standard Deviation = 1.069, why?

• We are dividing by (n-1)

### Standard Deviation

• Calculate the mean, range, and sample standard deviation for 2 units each at 4, 6, 8 and 10:

• Mean = 7

• Range = 6

• Sample Standard deviation = 2.39 and

Population Standard Deviation = 2.236, why not 2?

• Even though the data is an average of 2 units from the mean, the standard deviations are not exactly 2 because we are working with the square of the distances.

### Standard Deviation Summary

• Standard deviation is the square root of variance

• Population standard deviation has an “n” in the denominator

• Sample standard deviation has an “n – 1” in the denominator

• Both standard deviations is a measure of the spread of data

• The more the spread, the larger the standard deviation

### Class Work

• Pg 79, # 16, 18, 24

### Homework

• Page 78, #

• 5 – 9 all,

• 13 – 21 odd,

• 22

• 25 & 26

• Total of 13 problems