1. Measures of central tendency: Mode, median, mid-range and mean

2. The mode: – the most common (i) The number of errors made by 11 students: 4 6 7 5 9 10 7 6 4 7 8 Rearranging the numbers in order 4 4 5 6 6 7 7 7 8 9 10 The mode is 7 What type of data is this ? “This is discrete and ungrouped data ”

3. The mode: – the most common (ii) The number of children in 25 families: (grouped data) Number of children: 0 1 2 3 4 5 Number of families: 3 5 9 4 3 1

4. The mode: – the most common (ii) The number of children in 25 families: (grouped data) Number of children: 0 1 2 3 4 5 Number of families: 3 5 9 4 3 1 What is the mode? There are 9 families with 2 students so the mode is 2 (N.B not 9!)

5. The mode: – the most common (iii) Heights of 50 female students What type of data is this? (grouped continuous data) Height h cmFrequency 157 < h  159 4 159 < h  16111 161 < h  16319 163 < h  165 8 165 < h  167 5 167 < h  169 3 Total50 What is the mode??

6. The mode: – the most common (iii) Heights of 50 female students (grouped continuous data) Height h cmFrequency 157 < h  159 4 159 < h  16111 161 < h  16319 163 < h  165 8 165 < h  167 5 167 < h  169 3 Total50 The modal class is 161 < h  163 with 19 students 161 and 163 are the lower and upper class boundaries

7. 2. The median – the middle value 1.The number of errors made by 11 students: 4 6 7 5 9 10 7 6 4 7 8

8. 2. The median – the middle value (i).The number of errors made by 11 students: 4 6 7 5 9 10 7 6 4 7 8 Rearranging the numbers in order 4 4 5 6 6 7 7 7 8 9 10 Number of items n = 11 median is (n+1)/2 th value = 6 th value = 7

9. 2. The median – the middle value (i)If another student is added who made 5 mistakes 6 7 5 9 10 7 6 4 7 8 4 5 Rearranging the numbers in order 4 4 5 5 6 6 7 7 7 8 9 10 Number of items n = 12 Median is (n+1)/2 th value = 6.5 th value = (6+7)/2 = 6.5

10. 2. The median – the middle value (ii) Number of children in 25 families(grouped data) Number of children : 0 1 2 3 4 5 Number of families (freq): 3 5 9 4 3 1

11. 2. The median – the middle value (ii) Number of children in 25 families(grouped data) Number of children : 0 1 2 3 4 5 Number of families (freq): 3 5 9 4 3 1 Cumulative frequency : 3 8 17 21 24 25 Median value is (25 + 1)/2 th = 13 th value = 9 (values 9 to 17 are all 2) Cum Freq is a running total

12. 2. The median – the middle value (iii) Heights of 50 students (Grouped continuous data) Height h cmFrequency f 157 < h  159 4 159 < h  161 11 161 < h  163 19 163 < h  165 8 165 < h  167 5 167 < h  169 3 totals 50

13. 2. The median – the middle value Median class contains ½( 50 + 1)=25.5 th item. Median class is161 < h  163 Height h cmFrequency fCumulative freq 157 < h  159 44 159 < h  161 1115 161 < h  163 1934 163 < h  165 842 165 < h  167 547 167 < h  169 350 totals50

14. The mid-range -the value mid-way between the lowest and highest values (i) Student errors First re-arrange the data in numerical order 4 4 5 6 6 7 7 7 8 9 10 Mid-range is = (4 + 10)/2 =7

15. The mean (or average): add up all the values and divide by the number of values A population is a collection of items (usually a large number of values); the mean of the population is  A sample is a selection of the population; the mean of the sample is This is just a notation issue not really a big deal !

16. The mean (or average) (i) Student errors: 4 6 7 5 9 10 6 7 4 7 8  = = = 6.64 to 3 s.f.

17. or if we take a sample of just 4 students, say 7 9 6 4 Then == 6.5

18. The mean (or average) Number of children in 25 families(grouped data) For grouped data where f is the frequency

19. The mean (or average) ii) Number of children in 25 families(grouped data) Number of children (x): 0 1 2 3 4 5 Number of families (f) : 3 5 9 4 3 1 (xf): 0 5 18 12 12 5 = 2.08

20. The mean (or average) Heights of female students (grouped continuous data) Height h cmf xi fxi f 157 < h  159 4632 159 < h  161 111760 161 < h  163 193078 163 < h  165 81312 165 < h  167 5830 167 < h  169 3504 totals508116 = 162.32 cm What value of x doe we use here? Mid-point x i 160 162 164 166 168 158