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.

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2.4 Measures of Variation

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2 4 measures of variation

2.4 Measures of Variation

The Range of a data set is simply:

Range = (Max. entry) – (Min. entry)


Deviation

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

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

Population Standard Deviation

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


Sample 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

Sample Standard Deviation

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


Example

Example

  • Find the standard deviation of the following sample:


Example1

Example

  • Find the standard deviation of the following sample:


Example2

Example

  • Find the standard deviation of the following sample:


Example3

Example

  • Find the standard deviation of the following sample:


Example4

Example

  • Find the standard deviation of the following sample:

What will be the sum of this column?


Example5

Example

  • Find the standard deviation of the following sample:

What will be the sum of this column?

It will always be zero


Example6

Example

  • Find the standard deviation of the following sample:


Example7

Example

  • Find the standard deviation of the following sample:


Standard deviation

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 deviation1

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 deviation2

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 deviation3

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 deviation4

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 deviation5

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 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 curve from http allpsych com researchmethods images normalcurve gif

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


2 4 measures of variation

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


Class work

Class Work

  • Pg 79, # 16, 18, 24


Homework

Homework

  • Page 78, #

    • 5 – 9 all,

    • 13 – 21 odd,

    • 22

    • 25 & 26

    • Total of 13 problems


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