A class took a quiz. The class was divided into teams of 10 A class took a quiz. The class was divided into teams of 10. and each class had a mean average score of 81.5 Since the averages are the same, can we assume that the students in both classes all did pretty much the same on the exam?

Class A Mean Class B

NO The mean does not tell us anything about the grade distribution or variation of grades in the population Need a way to measure the spread of grades

Mean and range tell only part of the ‘story’ Mean average Mean and range tell only part of the ‘story’ Range (spread) Here, most of the numbers hover around the mean, and are not evenly distributed throughout the range.

72 - 81.5 = mean -9.5 Standard Deviation is a measure of how spread out the values in the data set are from the mean

SD is useful for making comparisons between data sets beyond visual impressions looks like a difference, but is it? In biology, to describe a population In education to calculate a grade curve In production systems as the upper and lower control limit; anything above or below represents a problem. In finance, the higher the standard deviation, the riskier the investment For sports teams, a high standard deviation shows that they perform well in some situations but not in others. Regarding climate while two cities may each have the same average maximum temperature, the standard deviation of the daily maximum temperature for a coastal city will be less than that of an inland city

SD describes a population If SD is small , the data is closer to the mean. More likely the IV is affecting the DV If SD is LARGE, the numbers are spread out from the mean. Other factors likely influencing the DV In biology, to describe a population In education to calculate a grade curve In production systems as the upper and lower control limit; anything above or below represents a problem. In finance, the higher the standard deviation, the riskier the investment For sports teams, a high standard deviation shows that they perform well in some situations but not in others. Regarding climate while two cities may each have the same average maximum temperature, the standard deviation of the daily maximum temperature for a coastal city will be less than that of an inland city SD describes a population

The Normal (Bell) curve F D C B A mean +/- 1 standard deviation = 68.3% of the group +/- 2 SD = 95.4% of the group

SD = the difference of all of the values from the mean of the population, summed, divided by the population size N-1? The standard deviation of a sample of a population uses 1/(N-1), generating an unbiased estimate. In the rare case when an entire population is counted/measured, the standard deviation calculation uses 1/N. ]

Here are the scores on the math quiz for team A: 72 76 80 81 83 84 85 89 Here are the scores on the math quiz for team A: ranked low to high mean = 81.5

1. Find all the distances from the mean (81.5) Distance from Mean 72 76 80 81 83 84 85 89 -9.5 - 5.5 - 1.5 0.5 1.5 2.5 3.5 7.5 To find the standard deviation … 1. Find all the distances from the mean (81.5)

2. Square each of the distances (to turn them into positive numbers) score d from Mean d2 72 76 80 81 83 84 85 89 - 9.52 = - 5.5 - 1.5 - 0.5 1.5 2.5 3.5 7.5 90.25 30.25 2.25 0.25 6.25 12.25 56.25 2. Square each of the distances (to turn them into positive numbers)

3. Add up all the squares Sum: 214.5 score d from Mean d2 72 76 80 81 83 84 85 89 - 9.5 - 5.5 - 1.5 - 0.5 1.5 2.5 3.5 7.5 90.25 30.25 2.25 0.25 6.25 12.25 56.25 Sum: 214.5 3. Add up all the squares

4. Divide by (n - 1) where n represents the total amount of numbers score d from Mean d2 90.25 30.25 2.25 0.25 6.25 12.25 56.25 72 76 80 81 83 84 85 89 - 9.5 - 5.5 - 1.5 - 0.5 1.5 2.5 3.5 7.5 Sum: 214.5 (10-1) = 23.8 4. Divide by (n - 1) where n represents the total amount of numbers

5. Take the square root of the average distance score d from Mean d2 90.25 30.25 2.25 0.25 6.25 12.25 56.25 72 76 80 81 83 84 85 89 - 9.5 - 5.5 - 1.5 - 0.5 1.5 2.5 3.5 7.5 Sum: 214.5 ______ 10-1 = 23.8 the variance SD = 4.88 5. Take the square root of the average distance 4.88 is the SD for this group

The standard deviation for Team B’s grades d from Mean d2 57 65 83 94 95 96 98 93 71 63 - 24.5 - 16.5 1.5 12.5 13.5 14.5 16.5 11.5 - 10.5 -18.5 600.25 272.25 2.25 156.25 182.25 210.25 132.25 110.25 342.25 Sum: 2280.5 (10 - 1) = 253.4 variance SD = 15.91 The standard deviation for Team B’s grades

Comparing the two classes Team A Team B Average on the Quiz Standard Deviation 81.5 81.5 4.88 15.91 On Team A, 68% of the scores (7 of 10) were between 77 and 86, compared to Team B where 68% of the students (7 of 10) scored between 65 and 97. Verify by looking at the original distribution of each set of scores! (slide 2)

TEAM A TEAM B 77 86 65 97 81.5 +/- 4.88 vs 81.5 +/- 15.91

TRY IT YOURSELF: A wildlife biologist measures the weight at birth of each of 5 pups in a litter of wolves. He finds the following: gender birth weight in kg M 2.2 M 2.8 F 1.8 F 2.4 F 2.5 Find the mean, variance, and standard deviation of the litter