Measures of central tendency Mode median midrange and

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

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

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

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!)

The mode: – the most common (iii) Heights of 50 female students What type of data is this? (grouped continuous data) Height h cm 157 < h 159 < h 161 < h 163 < h 165 < h 167 < h 169 Total Frequency 4 11 What is the mode? ? 19 8 5 3 50

The mode: – the most common (iii) Heights of 50 female students (grouped continuous data) Height h cm Frequency 157 < h 159 4 159 < h 161 11 161 < h 163 19 163 < h 165 8 165 < h 167 5 167 < h 169 3 Total 50 The modal class is 161 < h 163 with 19 students 161 and 163 are the lower and upper class boundaries

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

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 Number of items n = 11 median is (n+1)/2 th value = 6 th value =7 9 10

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 Number of items n = 12 Median is (n+1)/2 th value 8 = 6. 5 th value = (6+7)/2 = 6. 5 9 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

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

2. The median – the middle value (iii) Heights of 50 students (Grouped continuous data) Height h cm Frequency 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

2. The median – the middle value Median class contains ½( 50 + 1)=25. 5 th item. Median class is 161 < h 163 Height h cm 157 < h 159 < h 161 < h 163 < h 165 < h 167 < h 169 totals Frequency f 4 11 19 8 5 3 50 Cumulative freq 4 15 34 42 47 50

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

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 !

The mean (or average) (i) Student errors: 4 6 7 5 9 = = = = 6. 64 to 3 s. f. 10 6 7 4 7 8

or if we take a sample of just 4 students, say 7 9 6 4 Then = = 6. 5

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

The mean (or average) ii) Number of children in 25 families(grouped data) Number of children (x): 0 Number of families (f) : 3 (xf): 0 = 2. 08 1 5 5 2 9 18 3 4 12 4 3 12 5 1 5

The mean (or average) Heights of female students (grouped continuous data) f Mid-point xi 4 158 11 160 19 162 163 < h 165 8 164 165 < h 167 5 166 167 < h 169 3 168 xi f 632 1760 3078 totals 8116 Height h cm 157 < h 159 < h 161 < h 163 50 What value of x doe we use here? 1312 830 504 = 162. 32 cm