Unit 3 Section 3-3 – Day 1

3-3: Measures of Variation  Range – the highest value minus the lowest value.  The symbol R is used for range.  Variance – the average of the squares of the distance each value is from the mean.  The symbol is σ 2  Standard Deviation – the square root for the variance.  The symbol is σ

The Empirical Rule (Normal Rule) Section 3-3  Applies when the distribution is bell- shaped (or what is called normal ). Approximately 68% of the data falls within 1 standard deviation of the mean. Approximately 95% of the data falls within 2 standard deviation of the mean. Approximately 99.7% of the data falls within 3 standard deviation of the mean.

Empirical Rule Section 3-3

Finding the Variance (Population) Section 3-3  The formula for finding the variance of a population is: X = individual value μ = population mean N = population size

Finding the Variance (Sample) Section 3-3  The formula for finding the variance of a population is: X = individual value n = sample size

Steps for Finding Variance Section 3-3  Find the mean.  Find the difference between each data value and the mean.  Square each difference.  Find the sum of their squares.  Divide the sum by the number of data entries.

Finding the Variance Section 3-3  Find the variance for the population below: Value X-μ(X-μ)

Finding the Variance Section 3-3  Find the variance for the amount of European auto sales for a sample of 6 years as shown. Value X-μ(X-μ)

Steps for Finding Standard Deviation Section 3-3  First, determine the variance of the data set.  Then, take the square root of the variance.

Finding the Standard Deviation Section 3-2  Find the standard deviation for both of our previous examples.

Finding the Standard Deviation for Grouped Data Section 3-3  Formula:

Finding the Variance and Standard Deviation for Grouped Data Section 3-3  Make a table as shown  Find the midpoints of each class and place them in column C.  Multiply the frequency by the midpoint for each class, and place the product in column D.  Multiply the frequency by the square of the midpoint for each class, and place the product in column E.  Find the sums of column B, D, and E. ABCDE ClassFrequency f Midpoint X m f*X m f*X m 2

 To find the Variance:  Take the Sum of E, subtract away the quantity of the Sum of D squared divided by the Sum of B.  Then, divide your value by the Sum of B minus one.  To find the Standard Deviation, take the square root of the variance. Section 3-3

Finding the Variance and Standard Deviation: Grouped Data Section 3-3  Find the variance and standard deviation for the grouped data below: ClassFrequency f Midpoint X m f* X m f* X m – – – – – – – 40.52

Uses of the Variance and Standard Deviation  Variances and standard deviations can be used to determine the spread of data.  If they are large, the data is more dispersed  Useful in comparing two or more data sets to determine which is more variable.  Variances and standard deviations are used to determine the consistency of a variable.  Example: manufacturing nuts and bolts, the variation in diameters must be low so the parts fir together.  Variances and standard deviations are used to determine the number of data values that fall within a specific interval in a distribution.  Variance and standard deviations are used quite often in inferential statistics. Section 3-3

 Coefficient of Variation – the standard deviation divided by the mean.  Notation : CVar  The result is expressed as a percentage. Example : The mean of the number of sales of cars over a 3-month period is 87, and the standard deviation is 5. The mean of the commissions is $5225, and the standard deviation is $773. Compare the variations of the two. Section 3-3

 Range Rule of Thumb  A rough estimate of the standard deviation.  Standard deviation is approximately the range divided by four.  Chebyshev’s theorem – the proportion of values from a data set that will fall within k standard deviations of the mean will be at least 1-1/k 2, where k is a number greater than 1.  This can be applied to any data set regardless of its distribution or shape  Also states that three-fourths (or 75%) of the data values will fall within 2 standard deviations of the mean.

Chebyshev’s theorem Section 3-3  A survey of local companies found that the mean amount of travel allowance for executives was $0.25 per mile. The standard deviation was $0.02. Using Chebyshev’s theorem, find the minimum percentage of the data values that will fall between $0.20 and $0.30. Step 1 : Subtract the mean from the larger value Step 2 : Divide the difference by the standard deviation (find k) Step 3 : Use Chebyshev’s theorem to find the %

Homework  Pg : 18, 21, 31, 32, 35 Section 3-3