1. Minds on! Two students are being considered for a bursary. Sal’s marks are 97 92 84 71 Val’s marks are 95 90 86 73 Which student would you award the bursary to? Justify your choice.

2. 3.3 Measures of Spread Chapter 3 - Tools for Analyzing Data Learning goal: Calculate and interpret measures of spread (4) Questions? p. 159 #4, 5, 6, 8, 10-13 MSIP / Home Learning: p. 168 #2b, 3b, 4, 6, 7, 10 What is more important: potential or consistency?

3. What is spread? Measures of central tendency do not always tell you everything! These histograms have identical mean and median, but the spread is different! Spread is how closely the values cluster around the middle value

4. Why worry about spread? Less spread  greater confidence that values will fall within a particular range Important for making predictions

5. Measures of Spread We will study 4 Measures of Spread:  Range  Interquartile Range (IQR)  Variance / Standard Deviation (Std. Dev.) All 4 measure how spread out data is  Smaller value = less spread (more consistent)  Larger value = more spread (less consistent)

6. Measures of Spread 1) Range = (max) – (min)  Indicates the size of the interval that contains 100% of the data 2) Interquartile Range  IQR = Q3 – Q1, where  Q1 is the lower half median  Q3 is the upper half median  Indicates the size of the interval that contains the middle 50% of the data

7. Quartiles Example – 15 data values 26 28 34 36 38 38 40 41 41 44 45 46 51 54 55 Q2 = 41Median Q1 = 36Lower half median Q3 = 46Upper half median IQR = Q3 – Q1 = 46 – 36 = 10 So the middle 50% of the data is within a span of 10 units

8. Quartiles Example – 14 data values 26 28 34 36 38 38 40 | 41 41 44 45 46 51 54 If a quartile occurs between 2 values, it is calculated as the average of the two values  Q2 = 40.5 Q1 = 36 Q3 = 45 IQR = Q3 – Q1 = 45 - 36 = 9 The middle 50% of the data is within 9 units  This data is more consistent.

9. Box (and Whisker) Plot Min = 26 Q1 = 36Lower half median Q2 = 41Median Q3 = 46Upper half median Max = 55 26 36 41 46 55 This is one of the graph types in Fathom / Excel 2016 – you can hack Excel 2013 (see website).

10. A More Useful Measure of Spread Range is very basic  Does not take clusters nor outliers into account Interquartile Range is somewhat useful  Takes clusters into account  Visual in Box-and-Whisker Plot Standard deviation is very useful  Average distance from the mean for all data points

11. Deviation The mean of the numbers below is 48 deviation = (data) – (mean) The deviation for 24 is 24 - 48 = -24 -24 12 243648607284 36 The deviation for 84 is 84 - 48 = 36

12. Calculating Standard Deviation (σ) 1. Find the mean (average) 2. Find the deviation for each data point data point – mean 3. Square the deviations (data point – mean) 2 4. Average the squares of the deviations (this is called the variance, σ 2 ) 5. Take the square root of the variance

13. Example of Standard Deviation 26 28 34 36 mean = (26 + 28 + 34 + 36) ÷ 4 = 31 σ² = (26–31)² + (28-31)² + (34-31)² + (36-31)² 4 σ² = 25 + 9 + 9 + 25 4 σ² = 17 σ = √17 = 4.1 (1 dec. pl.)

14. Standard Deviation σ² (lower case sigma squared) is used to represent variance σ is used to represent standard deviation σ is commonly used to measure the spread of data, with larger values of σ indicating greater spread we are using a population standard deviation (next slide)

15. Different Types of Std. Dev. Population Standard Deviation Sample Standard Deviation

16. Different Types of Std. Dev. Standard Deviation Standard Deviation with Grouped Data

17. grouped mean = (2×2 + 3×6 + 4×6 + 5×2) / 16 = 3.5 deviations:  2: 2 – 3.5 = -1.5  3: 3 – 3.5 = -0.5  4: 4 – 3.5 = 0.5  5: 5 – 3.5 = 1.5 σ² = 2(-1.5)² + 6(-0.5)² + 6(0.5)² + 2(1.5)² 16 σ² = 0.7499 σ = √0.7499 = 0.9 Hours of TV 2345 Frequency2662

18. Measures of Spread in Excel Range  = max (data) – min (data) IQR  = quartile (data, 3) – quartile (data, 1) Population Standard Deviation  = stdev.p (data)

19. Measure of Spread - Recap Measures of Spread indicate how spread out data is Smaller value means data is more consistent 1) Range = Max – Min 2) Interquartile Range: IQR = Q3 – Q1, where  Q1 = first half median  Q3 = second half median 3) Standard Deviation i. Find mean (average) ii. Find all deviations (data point) – (mean) iii. Square all iv. avg them (this is variance or σ 2 ) v. Take the square root to get std. dev., σ

20. MSIP / Home Learning Read through the examples on pp. 164-167 Complete p. 168 #2b, 3b, 4, 6, 7, 10 You are responsible for knowing how to do simple examples by hand (~6 pieces of data) We will use technology (Fathom/Excel) to calculate larger examples Have a look at your calculator and see if you have this feature (Σσn and Σσn-1)

22. 3.4 Normal Distribution Chapter 3 – Tools for Analyzing Data Learning goal: Determine the % of data within intervals of a Normal Distribution Questions?p. 168 #2b, 3b, 4, 6, 7, 10 MSIP / Home Learning: p. 176 #1, 3b, 6, 8-10 “Think of how stupid the average person is, and realize half of them are stupider than that.” -George Carlin

23. Histograms Histograms can be skewed... Right-skewed Left-skewed CD Collection Roll of coins

24. Histograms... or symmetrical

25. Normal? A Normal distribution is a histogram that is symmetrical and has a bell shape It is used quite a bit in statistical analysis Also called a Gaussian Distribution Symmetrical with equal mean, median and mode that fall on the line of symmetry of the curve

26. A Real Example the heights of 600 randomly chosen Canadian students from the “Census at School” data set the data approximates a Normal distribution

27. The 68-95-99.7% Rule Area under curve is 1 (i.e. it represents 100% of the data) Approx 68% of the data falls within 1 standard deviation of the mean Approx 95% of the data falls within 2 standard deviations of the mean Approx 99.7% of the data falls within 3 standard deviations of the mean http://davidmlane.com/hyperstat/A25329.html

28. Distribution of Data 34% 13.5% 2.35% 68% 95% 99.7% xx + 1σx + 2σx + 3σx - 1σx - 2σx - 3σ 0.15%

29. Normal Distribution Notation The notation above is used to describe the Normal distribution where x is the mean and σ² is the variance (square of the standard deviation) e.g., X~N (74,8 2 ) describes a Normal distribution with mean 74 and standard deviation 8 (e.g., class marks)

30. Example 1a) The time before burnout for a brand of LED averages 120 months with a standard deviation of 10 months and is approximately Normally distributed. So X~N(120,10 2 ). What is the length of time a user might expect an LED to last with: a) 68% confidence? b) 95% confidence?

31. Example 1) continued… 34% 13.5% 2.35% 95% 99.7% 12014015010090 months 68% 110 months 130 months

32. Example 1) cont’d 68% of the data will be within 1 standard deviation of the mean This will mean that 68% of the bulbs will be between 120–10 = 110 months and 120+10 = 130 months So 68% of the bulbs will last 110 - 130 months 95% of the data will be within 2 standard deviations of the mean This will mean that 95% of the bulbs will be between 120 – 2×10 = 100 months and 120 + 2×10 = 140 months So 95% of the bulbs will last 100 - 140 months

33. Example 1b) Suppose you wanted to know how long 99.7% of the bulbs will last This is the area covering 3 standard deviations on either side of the mean This will mean that 99.7% of the bulbs will be between 120 – 3×10 months and 120 + 3×10 So 99.7% of the bulbs will last 90-150 months This assumes that all the bulbs are produced to the same standard

34. Example 1c) What % of LEDs will last between  90 and 130 months?  120-140 months?

35. Example 1c 34% 13.5% 2.35% 95% 99.7% 12014015010090 months What % of LEDs will last between 90 and 130 months? 2.35+13.5+34+34 = 88.85% What % of LEDs will last between 110 and 140 months? 34+34+13.5=81.5 110 months 130 months

36. Percentage of data between two values The area under any normal curve is 1 The percent of data that lies between two values in a normal distribution is equivalent to the area under the normal curve between these values See examples 2 and 3 on page 175

37. Why is the Normal distribution so important? Many psychological and educational variables are distributed Normally  height, memory, IQ, reading ability, etc. Normal distributions are statistically easy to work with  All kinds of statistical tests are based on them Lane (2003)

38. MSIP / Home Learning Complete p. 176 #1, 3b, 6, 8-10 http://onlinestatbook.com/

39. References Lane, D. (2003). What's so important about the normal distribution? Retrieved October 5, 2004 from http://davidmlane.com/hyperstat/normal_distri bution.html Wikipedia (2004). Online Encyclopedia. Retrieved September 1, 2004 from http://en.wikipedia.org/wiki/Main_Page