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: X Sum

Example Find the standard deviation of the following sample: X Sum

Example Find the standard deviation of the following sample: X Sum

Example Find the standard deviation of the following sample: X Sum54

Example Find the standard deviation of the following sample: X Sum54 What will be the sum of this column?

Example Find the standard deviation of the following sample: X Sum540 What will be the sum of this column? It will always be zero

Example Find the standard deviation of the following sample: X Sum54

Example Find the standard deviation of the following sample: X Sum5415.5

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 Curve from

Standard Deviation in a Normal Curve from

Class Work Pg 79, # 16, 18, 24

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