3-2 Measures of Variance

Symbols n: sample size N: population size ∑: sum f: frequency : sample mean µ: population mean s 2: sample variance σ2: population variance s : sample standard deviation σ: population standard deviation Quiz tomorrow on these!

Remember: Section 3-1 was about measures of central tendency Mean () Median mode

Measures of Variation: Range Variance (2) Standard Deviation ()

1. Range: maximum - minimum Measures of Variation 1. Range: maximum - minimum Advantage: quick and easy Disadvantage: only considers 2 numbers

Measures of Variation 2. Variance Describes the variation in the data Units are usually meaningless

Measures of Variation 3.Standard Deviation Describes the variation or average distance the data is from the mean Gives the “average deviation” Units are meaningful

Formulas Population Standard Deviation Sample Standard Deviation *Beware: We must now start determining whether a problem is a sample or population.

Example 1: Here’s how the standard deviation formula works: We will use this example. Mrs. Sigette asked 5 people at the game how much money they had in their pocket. Those people had $70, $75, $80, $85, $90. The mean is $80. The range is $20.

How much does each data point differ (or deviate) from the mean? x $70 70 – 80 = -10 $75 75 – 80 = -5 $80 80 – 80 = 0 $85 85 – 80 = 5 $90 90 – 80 = 10

Since we are looking for the average of that difference, we need to add them up. The problem is they add up to zero, so mathematicians decided to eliminate the negatives by squaring all of the differences (easier than taking the absolute value). x $70 -10 $75 -5 $80 $85 5 $90 10  = 0 100 25  = 250

The variance (σ2) is 50 square dollars. To get the standard deviation or average of the deviations or variance, you must add all of the deviations and divide by the number of data points you have. The variance (σ2) is 50 square dollars. Whaaaaat??

This number is actually the variance, but it is not in a useful unit (square dollars?). Therefore to put it back into useful units, we will take the square root of it.

Stock A and B Range = ? Range = ? Mean = 61.4 Mean = 61.5 Standard Deviation = ? Range = ? Mean = 61.5 Standard Deviation = ?

Example 2: Value of Stock A 56 57 58 61 63 67 Range: Mean ( ): n: Sample Variance: s2 = Standard Deviation: s =

Example 2: Value of Stock B 56 75 48 52 57 67 33 77 82 = Range: Mean ( ): n: Sample Variance: s2 = Standard Deviation: s =

Stock A and B Range = Mean = 61.5 Standard Deviation = Range =

Starting salaries for Corporation X (1000s of dollars) Example 3: x (x-)2 40 23 41 50 49 32 29 52 58 = Range: Mean ( ): N: Population Variance: 2 = Standard Deviation:  =