1. 3.3 Measures of Spread Chapter 3 - Tools for Analyzing Data Learning goal: calculate and interpret measures of spread Due now: 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?

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

3. Why worry about spread? Less spread means you have greater confidence that values will fall within a particular range Important for making predictions

4. Measures of Spread We will also study 3 Measures of Spread:  Range  Interquartile Range (IQR)  Standard Deviation (Std.Dev.) All 3 measure how spread out data is  Smaller value = less spread (more consistent)  Larger value = more spread (less consistent)

5. 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

6. Quartiles Example 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 The middle 50% of the data is within 10 units If a quartile occurs between 2 values, it is calculated as the average of the two values

7. 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

8. Deviation The mean of these numbers 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

9. Calculating Standard Deviation 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) 5. Take the square root of the variance

10. 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

11. AGENDA Recap of Measures of Spread The Standard Deviation ‘formula’ Measures of Spread with Grouped Data Measures of Spread in Excel Measures of Spread – 7 species

12. Measure of Spread - Recap Measures of Spread are numbers indicating how spread out data is Smaller value for any measure of spread 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./iv. Square all and avg them (this is variance or σ 2 ) v. Take the square root to get std. dev. σ

13. 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

14. Different Types of Std. Dev.

15. Measures of Spread in Excel Range  = max (data) – min (data) IQR  =quartile (data, 3) – quartile (data, 1) Standard Deviation  =stdevp (data)

16. Standard Deviation with Grouped Data 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

17. MSIP / Home Learning Read through the examples on pages 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)

18. 3.4 Normal Distribution Chapter 3 – Tools for Analyzing Data Learning goal: Determine the % of data within intervals of a Normal Distribution Due now: 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

19. Histograms Histograms can be skewed... Right-skewed Left-skewed CD Collection Year on Coins

20. Histograms... or symmetrical

21. Normal? A normal distribution is a histogram that is symmetrical and has a bell shape 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

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

23. The 68-95-99.7% Rule Area under curve is 1 (i.e. it represents 100% of the population surveyed) 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

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

25. 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 (70,8 2 ) describes a Normal distribution with mean 70 and standard deviation 8 (our class at midterm?)

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

27. An example 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

28. Example continued… 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

29. Example continued… 34% 13.5% 2.35% 95% 99.7% 12014015010090 months

30. 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

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

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

33. 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