2.4 Measures of Variation

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# 2.4 Measures of Variation - PowerPoint PPT Presentation

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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Presentation Transcript

### 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
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