Chapter 9 Statistics

1. Chapter 9Statistics Section 9.2 Measures of Variation

2. Who Has Better Scores? • Adam and Bonnie are comparing their quiz scores in an effort to determine who is the “best”. Help them decide by calculating the mean, median, and mode for each. Adam’s ScoresBonnie’s Scores 85 81 60 85 105 86 85 85 72 90 100 80

3. The Winner? So, who has the better quiz scores?

4. Another Way to Compare • Sometimes the measures of central tendency (mean, median, mode) aren’t enough to adequately describe the data. • We also need to take into account the consistency, or spread, of the data.

5. Range • The range of the data is the difference between the largest and smallest number in a sample. • Find the range of Adam and Bonnie’s scores. • Adam: 105 – 60 = 45 • Bonnie: 90 – 80 = 10 • Based on the range, Bonnie’s scores are more consisent, and some might argue therefore, better than Adam’s.

6. Another Measure of Dispersion • The most useful measure of variation (spread) is the standard deviation. • First, we will look at the deviations from the mean.

7. Deviations from the Mean • The deviation from the mean is the difference between a single data point and the calculated mean of the data. • Data point close to mean: small deviation • Data point far from mean: large deviation • Sum of deviations from mean is always zero. • Mean of the deviations is always zero.

8. Deviations from Mean for Adam and Bonnie Adam Bonnie

9. Variance • Because the average of the mean deviation is always zero, we must modify our approach using the variance. • The variance is the mean of the squares of the deviation.

10. Variance for Adam and Bonnie’s Scores • Using the deviations from the mean we have already calculated for Adam and Bonnie, we will find the variance for each. • Adam : s² = • Bonnie: s² = 1417.5 5 = 283.5 s² = 65.5 5 = 13.1 s² =

11. Standard Deviation • To find the variance, we squared the deviations from the mean, so the variance is in squared units. • To return to the same units as the data, we use the square root of the variance, the standard deviation. Variance Standard Deviation

12. Adam and Bonnie’s Standard Deviation • Adam: • Bonnie: • Based on the standard deviation, Bonnie’s scores are better because there is less dispersion. In other words, she is more consistent than Adam.

13. Formulas for Variance and Standard Deviation

14. Example 1 • The number of homicide victims in Vermont from 1992 through 2001 is given in the table at right. (Source: http://170.222.24.9/cjs/crime_01/homicide_01.html) • Find the mean, median, mode, and standard deviation of the data.

15. Sample vs. Population • The mean, variance, and standard deviation of a random sample is referred to as the sample mean ( ), sample variance (s²), and sample standard deviation (s). • The sample mean, variance, standard deviation, etc. can only give us an approximation to the population mean (µ), the population variance (σ²), and the population standard deviation (σ). • The main difference lies in the denominator of the formulas for standard deviation. When the value of the sample, n, is large, the sample standard deviation gives a good estimate of the population standard deviation.

16. Grouped Distributions

17. Example 2 • Mr. Smith recently gave a math test and organized his scores into the table at right. Help Mr. Smith determine the class average, the median score, and standard deviation.

18. Chebyshev’s Theorem • Chebyshev’s Theorem states that for any set of numbers, the fraction (or probability) that will lie within k standard deviations of the mean (for k > 1) is at least __1__ k ² 1 -

19. Example 3 • Use Chebyshev’s Theorem to find the fraction of all the numbers of a data set that must lie within 4 standard deviations from the mean.

20. Example 4 • In a certain distribution of numbers, the mean is 50 with a standard deviation of 6. Use Chebyshev’s Theorem to tell the probability that a number lies in each interval. a.) between 38 and 62 b.) less than 38 or more than 62