Do Now Find the mean and standard deviation for the data set (assume sample data):

Population Variance (σ 2 ) Standard Deviation (σ) Sample Variance (s 2 ) Standard Deviation (s) The reason we use (n-1) for the sample variance and standard deviation is because using n does not give us the best estimate of the population variance. This is because when the population is large and the sample is small, the variance computed using n underestimates the population variance. Therefore we divide by (n-1), giving a slightly larger value and an unbiased estimate of population variance.

Variance and Standard Deviation for Grouped Data Procedure: 1. Make a table as shown and find the midpoint of each class. 2. Multiply the frequency by the midpoint for each class, and place the products in column D. 3. Multiply the frequency by the square of the midpoint and place the product in column E. 4. Find the sums of columns B, D, and E. 5. Substitute in the formula and solve for s 2 to get the variance. 6. Take the square root to get the standard deviation. Example: miles run per week ABCDE ClassFrequencyMidpointf  X m f  X 2 m

Coefficient of Variation Allows you to compare standard deviations when the units are different – such as comparing the number of sales per salesperson over a 3-month period and the commissions made by these salespeople. Standard deviation divided by the mean, expressed as a percentage. Denoted by Cvar Example: Homework Problem from last night

Range Rule of Thumb A rough estimate of standard deviation is s ≈ range 4 ***Only as approximation Should be used when the distribution of data values is unimodal and roughly symmetric. Can be used to estimate the largest and smallest data values of a data set. Approximately 2 standard deviations away from the mean Example – Cigarette Taxes

Chebyshev’s Theorem

Chebyshev’s Theorem Examples Example 1 The mean price of houses in a certain neighborhood is $50,000, and the standard deviation is $10,000. Find the price range for which at least 75% of the houses will sell. Example 2 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.

Empirical Rule Chebyshev’s Theorem applies to ANY distribution, regardless of shape. However, when a distribution is bell-shaped (or what is called normal), the following statements, which make up the empirical rule are true: Approximately 68% of the data values will fall within 1 standard deviation of the mean. Approximately 95% of the data values will fall within 2 standard deviation of the mean. Approximately 99.7% of the data values will fall within 3 standard deviation of the mean.

Empirical Rule Example Suppose that the scores on a national achievement exam have a mean of 480 and a standard deviation of 90. If these scores are normally distributed, then Approximately 68% will fall between ______ and _______, Approximately 95% will fall between ______ and _______, And approximately 99.7% will fall between ______ and ______.

Skills check – Chebyshev and Empirical 1.Apply Chebyshev’s Theorem to the systolic blood pressure of normotensive men. At least how many of the men in the study fall within 1 standard deviation of the mean? 2.At least how many of those men in the study fall within 2 standard deviations of the mean? 3.Give the ranges for the diastolic blood pressure (normotensive and hypertensive) of older women. Assume normal distribution. 4.Do the normotensive, male, systolic blood pressure ranges overlap with the hypertensive, male, systolic blood pressure ranges? Assume normal distribution. NormotensiveHypertensive MenWomenMenWomen (n=1200)(n=1400)(n=1100)(n=1300) Age55 ± ± 1064 ± 10 Blood Pressure (mm Hg) Systolic123 ± 9121 ± ± ± 20 Diastolic78 ± 776 ± 791 ± 1088 ± 10