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2.4 Measures of VariationPowerPoint Presentation

2.4 Measures of Variation

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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?

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

Standard Deviation in a Normal Curvefrom http://allpsych.com/researchmethods/images/normalcurve.gif

Standard Deviation in a Normal Curvefrom http://www.comfsm.fm/~dleeling/statistics/normal_curve_diff_sx.gif

Class Work

- Pg 79, # 16, 18, 24

Homework

- Page 78, #
- 5 – 9 all,
- 13 – 21 odd,
- 22
- 25 & 26
- Total of 13 problems

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