1. Measures of the Central of a Distribution: M ean –i–i s what we know as the average –s–s ymbol:, x bar –i–i s generally a good representative of the data –c–c an be influenced by extreme values

2. M edian –t–t he middle number (data must be ordered) –m–m edian of an odd number of data is one of the data –m–m edian of an even number of data is the mean of the two middle values n ot necessarily one of the data –i–i s not affected by extreme values

3. Mode – the value that occurs most frequently – data can be bimodal (2 modes) – data can have no mode – is not affected by extreme values – is always a member of the set of data Measures of the Central of a Distribution:

4. The mean of the ten numbers listed below is 5.5. 4, 3, a, 8, 7, 3, 9, 5, 8, 3 (a)Find the value of a. (b)Find the median of these numbers. 5 5

5. Find the mode. Finding mean, median and mode from a frequency table.

6. Find the mean. 1.Multiply the value by the frequency. 2.Add products. 3.Divide by the sum of the frequency. To find the mean: f ∙x Finding mean, median and mode from a frequency table. 6.95

7. Measures of Central Tendency – statistics that describe the tendency of data to center about certain numerical values. – mean – median – mode Measures of Variability – describes the spread of the data. – range – interquartile range – standard deviation Measures of the Spread of a Distribution:

8. {35, 40, 45}and{10, 40, 70} 40 is the mean of both sets, yet the variability is much greater in the second set than in the first. Compare:

9. Measures of Spread: Range Inter-quartile range Standard deviation the difference of the greatest and least values.

10. Range the difference of the greatest and least values. Range = Largest Value – Smallest Value The number of piglets in the litter of 10 pigs are: 10 12 12 13 15 16 9 10 14 11 Find the range. 16 – 9 = 7 Measures of the Spread of a Distribution:

11. Measures of Spread: Range Interquartile range Standard deviation the difference of the greatest and least values. IQR = Q 3 – Q 1 the range of the middle half of the data.

12. Consider a set of data: 2 3 3 3 4 4 5 5 5 5 6 6 6 7 7 8 9 Find the median (to cut the data in half). Find the median of each half. 3.56.5

13. The median is the second quartile, Q 2 or 50 th percentile The lower quartile, Q 1, is the median of the lower half of the data or 25 th percentile The upper quartile, Q 3, is the median of the upper half of the data or 75 th percentile The inter-quartile range is the difference in the upper quartile and the lower quartile. IQR = Q 3 – Q 1

14. Consider a set of data: 2 3 3 3 4 4 5 5 5 5 6 6 6 7 7 8 9 Find the median (to cut the data in half) Find the median of each half IQR = Q3 Q3 - Q1Q1 Half of the values are between 3.5 and 6.5. = 6.5 – 3.5 = 3

15. For the data set: 9, 8, 2, 3, 7, 6, 5, 4, 5, 4, 6, 8, 9, 5, 5, 5, 4, 6, 6, 8 Find the: a) median b) lower quartile c) upper quartile d) inter-quartile range 5.5 4.5 7.5 3

16. For the data set: 6, 4, 9, 15, 5, 13, 7, 12, 8, 10, 4, 1, 13, 1, 6, 4, 5, 2, 8, 2 Find the: a) median b) lower quartile c) upper quartile d) inter-quartile range

17. Standard Deviation The most widely used measure of the spread of a sample. Measures the deviation between data values and the mean. – The larger the standard deviation, the more widely spread the data (and vice versa). Measures of the Spread of a Distribution:

18. Standard Deviation x is any score is the mean n is the number of scores

19. Find the mean and standard deviation on the calculator: Type data in List 1 1-Var Stats L1 On paper you’ll see ‘s’ being used to standard for standard deviation. But you should use the σ measurement from the calculator.

20. Find the standard deviation of a frequency table on the calculator: Type data in L1 Type frequency in L2 1-Var Stats L1, L2

21. Find the mean and standard deviation on GDC: Find the mean and SD of this data set: 4 5 6 8 12 13 2 5 6 9 10 9 8 3 5 mean = 7 Standard deviation = 3.10

22. Find the standard deviation of a frequency table on GDC: Type data in L1 Type frequency in L2 1-Var Stats L1, L2

23. Find the standard deviation of the following distribution of scores. ScoreFrequency 113 124 135 142 158 169 175 188 196 s.d. = 2.38

24. Box Plots Box-and-whisker plots are useful ways to summarize data and illustrate its variability. It consists of a rectangular box with hinges at Q1 Q1 and Q3.Q3. Segments extending from the ends of the box are called whiskers which stop at the extreme values of the set. Box-and-Whisker Plots

25. Box Plots The inter-quartile range is the width of the box. The maximum length of each whisker is 1.5 times the inter- quartile range. Any data value that is larger than (or smaller than) 1.5 × IQR is marked as an outlier.

26. To Create a Box-and-Whisker Plot: 1)Make a number line. 2)Create the box between Q 1 and Q 3. 3)Draw in Q 2. 4)Determine any outliers: Upper boundary = Q 3 + 1.5(IQR) Lower boundary = Q 1 – 1.5(IQR) 5)Plot any outliers. 6)Extend the whiskers to the maximum & minimum (provided they’re not outliers).

27. Make a box-and-whisker plot from this data. 42 58 61 64 68 69 72 74 74 75 78 78 79 82 82 82 86 88 91 94 98 99 Test Scores, 3 rd Period Class Problem 1

28. To do a Box-and-Whisker on the Calculator: Type data into L1 Go to StatPlot and turn a plot on Choose the box-and-whisker with outliers Choose your list Zoom 9 : Stat

29. The Military Draft Lottery For more than 50 years, Selective Service and the registration requirement for America's young men have served as a backup system to provide manpower to the U.S. Armed Forces. President Franklin Roosevelt signed the Selective Training and Service Act of 1940 which created the country's first peacetime draft and formally established the Selective Service System as an independent Federal agency. From 1948 until 1973, during both peacetime and periods of conflict, men were drafted to fill vacancies in the armed forces which could not be filled through voluntary means. (Source: Selective Service System - June 25, 2001 revision) http://www.landscaper.net/draft.htm

30. The Military Draft Lottery In 1973, the draft ended and the U.S. converted to an All- Volunteer military. The registration requirement was suspended in April 1975. It was resumed again in 1980 by President Carter in response to the Soviet invasion of Afghanistan. Registration continues today as a hedge against underestimating the number of servicemen needed in a future crisis. (Source: Selective Service System - June 25, 2001 revision) http://www.landscaper.net/draft.htm

31. The Military Draft Lottery December 1, 1969 marked the date of the first draft lottery held since 1942. This drawing determined the order of induction for men born between January 1, 1944 and December 31, 1950. A large glass container held 366 blue plastic balls containing every possible birth date and affecting men between 18 and 26 years old. (Source: Selective Service System - June 25, 2001 revision) http://www.landscaper.net/draft.htm

32. The Military Draft Lottery The first date drawn from the glass container received draft number one and eligible men born on that date were drafted first. In a truly random lottery there should be no relationship between the date and the draft number.

33. 1970 Military Draft Lottery This dataset suggests that men born later in the year were more likely to be drafted.