1. Measures of Central Tendency Chapter 3.2 – Tools for Analyzing Data Mathematics of Data Management (Nelson) MDM 4U

2. Sigma Notation the sigma notation is used to compactly express a mathematical series ex: 1 + 2 + 3 + 4 + … + 15 this can be expressed: the variable k is called the index of summation. the number 1 is the lower limit and the number 15 is the upper limit we would say: “the sum of k for k = 1 to k = 15”

3. Example 1: write in expanded form: This is the sum of the term 2n+1 as n takes on the values from 4 to 7. = (2×4 + 1) + (2×5 + 1) + (2×6 + 1) + (2×7 + 1) = 9 + 11 + 13 + 15 = 48 NOTE: any letter can be used for the index of summation, though a, n, i, j, k & x are the most common

4. Example 2: write the following in sigma notation

5. The Mean Found by dividing the sum of all the data points by the number of elements of data Affected greatly by outliers Deviation  the distance of a data point from the mean  calculated by subtracting the mean from the value  i.e.

6. The Weighted Mean where x i represent the data points, w i represents the weight or the frequency “The sum of the products of each item and its weight divided by the sum of the weights” see examples on page 153 and 154 example: 7 students have a mark of 70 and 10 students have a mark of 80 mean = (70×7 + 80×10) ÷ (7+10) = 75.9

7. Means with grouped data for data that is already grouped into class intervals (assuming you do not have the original data), you must use the midpoint of each class to estimate the weighted mean see the example on page 154-5 and today’s Example 4

8. Median the midpoint of the data calculated by placing all the values in order if there is an odd number of values, the median is the middle number  1 4 6 8 9median = 6 if there are an even number of values, the median is the mean of the middle two numbers  1 4 6 8 9 12 median = 7 not affected greatly by outliers

9. Mode The number that occurs most often There may be no mode, one mode, two modes (bimodal), etc. Which distributions from yesterday have one mode? Mound-shaped, Left/Right-Skewed Two modes? U-Shaped, some Symmetric Modes are appropriate for discrete data or non-numerical data  Eye colour  Favourite Subject

10. Distributions and Central Tendancy the relationship between the three measures changes depending on the spread of the data symmetric (mound shaped)  mean = median = mode right skewed  mean > median > mode left skewed  mean < median < mode

11. What Method is Most Appropriate? Outliers are data points that are quite different from the other points Outliers affect the mean the greatest The median is least affected by outliers Skewed data is best represented by the median If symmetric either median or mean If not numeric or if the frequency is the most critical measure, use the mode

12. Example 3 a) Find the mean, median and mode mean = [(1x2) + (2x8) + (3x14) + (4x3)] / 27 = 2.7 median = 3 (27 data points, so #14 falls in bin 3) mode = 3 b) What shape does it have? Left-skewed Survey responses 1234 Frequency 28143

13. Example 4 Find the mean, median and mode mean = [(145.5 × 3) + (155.5 × 7) + (165.5 × 4)] ÷ 14 = 156.2 median = 151-160 or 155.5 mode = 151-160 or 155.5 MSIP / Homework: p. 159 #4, 5, 6, 8, 10-13 Height 141-150151-160161-170 No. of Students 374

14. MSIP / Homework p. 159 #4, 5, 6, 8, 10-13

15. References Wikipedia (2004). Online Encyclopedia. Retrieved September 1, 2004 from http://en.wikipedia.org/wiki/Main_Page