Statistics – Measure of Central Tendency, Mean , Mode and Median
Do you know that Singaporeans consumed around 665 tonnes of rice and 430 tonnes of seafood per day? The department of Statistics in Singapore collects and classifies various records such as these for administration reference so that policies and decisions can be better formulated to meet the needs of the country.
Last year, we learnt how to organize statistical date and present them in a diagram so that they can understand and important information can be extracted from the data statisticians also analyze them to deduce any useful information from the raw data. An important component of data analysis is to find the average value( mean ), the middle value ( median ) or the most frequent value ( mode ) to represent or describe a whole set of data.
The mean The mean of a set of numbers is the sum of numbers divided by the number of numbers in the set. Sum of the numbers Mean = Number of numbers
The median (a) The median for an odd number of numbers is the middle number when the numbers are arranged in order of increasing magnitude. (b) The median for an even number of numbers is the man of the two middle numbers when the numbers are arranged in order of increasing magnitude.
The mode The number which occurs most frequently in a set of numbers is called the mode of the set of the numbers.
Example 1 Find the mean, median and mode of the following sets of scores.
(a) 4, 7, 8, 3, 4, 8, 2, 8, 9
Sum of the numbers Mean = Number of numbers 4 + 7 + 8 + 3 + 4 + 8 + 2 + 8 + 9 = 9 53 = 9 = 5.8˙
** To find the median , we have to arrange the scores in ascending order.
4, 7, 8, 3, 4, 8, 2, 8, 9 2, 3, 4, 4, 7, 8, 8, 8, 9
Median = 7
Mode = 8 ( occurs most frequent ) (b) 33, 39, 29, 39, 36, 33, 30, 33, 35, 39
Sum of the numbers Mean = Number of numbers 33 + 39 + 29 + 39 + 36 + 33 + 30 + 33 + 35 + 39 = 10 346 = 10 = 34.6
** To find the median , we have to arrange the scores in ascending order.
33, 39, 29, 39, 36, 33, 30, 33, 35, 39 29, 30, 33, 33, 33, 35, 36, 39, 39, 39
Median 33 + 35 = 2 = 34
Mode = 33 and 39 ( since both have the same number of frequency )
Skill Practice 13A ( Part 1 ) : Do in Exercise Book A
1. Determine the mean, median and mode of the following sets of numbers.
1 10, 11, 13, 11, 15, 16, 11 2 63, 80, 54, 70, 51, 72, 64, 66 3 46, 78, 97, 45, 67, 99, 57, 46, 65 4 11.1, 10.1, 9.8, 9.7, 9.9, 11.1, 10.2, 10.1, 9.8, 9.6, 10.4, 9.8 5 2, 5, 6, 3, 7, 8, 4, 12, 11, 9, 10, 7, 6, 8, 9, 7
Example 2 Determine the mean, median and mode of the following table.
x 2 4 6 8 10 12 f 2 4 10 6 3 1
2 2 + 44+ 6 10 + 8 6 + 10 3 + 12 1 Mean = 2 + 4+ 10 + 6 + 3 + 1 4 + 16 + 60 + 48 + 30 + 12 = 26 170 = 26 = 6.54 ( 3 sig fig )
Median = 6 ( the mean of 13th and 14th term )
Mode = 6 Example 3 Two dice are tossed 31 times. The sum of the score each time is shown below :
Score (s) 2 3 4 5 6 7 8 9 10 11 12 Frequency (f) 1 1 3 4 6 8 3 2 1 1 1
2 1 + 3 1 + 4 3 + 5 4 + 6 6 + 7 8 + 9 2 + 10 1 + 11 1 + 12 1 Mean = 31 2 + 3 + 12 + 20 + 36 + 56 + 24 + 18 + 10 + 11 + 12 = 31 204 = 31 = 6.58 ( 3 sig fig )
Median = 7 ( the 16th score )
Mode = 7
Example 4
The following diagram illustrate the number of children per family of a sample of 100 families 45 in a certain housing estate. 40 35 (a) State the modal number of children per 30 s e
family. i l i 25 (b) Calculate the mean number of children m a f
f 20
per family. o
o
(c) Find the median number of children per N 15
family. 10
5
0 0 1 2 3 4 No of Children per fam ily
(a) The modal number of children per family is 2. (b) 5 0 + 20 1 + 40 2 + 25 3 + 10 4 Mean = 100 0 + 20 + 80 + 75 + 40 = 100 215 = 100 = 21.5 (c) Median number of children per family = 2 Example 5 The following are the marks scored by 40 students in a Physics test marked out of a total of 10. 8 6 4 3 5 5 2 9 2 7 9 3 3 7 7 5 8 3 7 3 4 8 7 8 2 4 6 2 4 1 7 7 6 2 6 4 4 6 10 6
(a) Draw a frequency table and a histogram for the data. (b) What is the average score? (c) What is the most frequent score? (d) What is the median score? (e) What is the range, that is, the difference between the highest and the lowest scores?
(a) Score(s) 1 2 3 4 5 6 7 8 9 10 Frequency (f) 1 5 5 6 3 6 7 4 2 1
8
) 7 f ( 6 y
c 5 n
e 4 u
q 3 e r 2 F 1 0 1 2 3 4 5 6 7 8 9 10 Score (s)
(b) Average score ( Mean ) 1� 1 � 2 � 5 � 3 � 5 � 4 � 6 � 5 � 3 6 6 7 7 8 4 9 2 10 1 = 40 1+ 10 + 15 + 24 + 15 + 36 + 49 + 32 + 18 + 10 = 40 210 = 40 = 5.25
(c) Mode = 7
(d) 5+ 6 = Median 2 = 5.5 ( the mean of the 20th and 21st score )
(e) The range = 10 – 1 = 9 Example 6 In a mathematics quiz, the scores obtained by 36 participants are : 45 41 52 68 75 84 99 41 45 48 54 48 59 58 54 68 68 53 53 53 65 60 70 78 71 73 80 81 91 81 93 83 85 87 94 96 (a) Draw a stem and leaf diagram. (b) Find the mean, mode and median.
(a) Stem Leaf 4 5 1 1 5 8 8 5 2 4 9 8 4 3 2 3 6 8 8 8 5 0 7 5 0 8 1 3 8 4 0 1 1 3 5 7 9 9 1 3 4 6
Rearrange Stem Leaf 4 1 1 5 5 8 8 5 2 2 3 3 4 4 8 9 6 0 5 8 8 8 7 0 1 3 5 8 8 0 1 1 3 4 5 7 9 1 3 4 6 9 * 5 2 stands for 52
Note : To draw the above stem and leaf diagram, firstly classify the scores into groups of forties, fifties,.. etc. List the ten digits in the column of “stem” and all the units digits are written in the same row according to the ten digit in the space of “leaf”.
(b) Subtotal of each rows : 1st row : 40 6 + (1 + 1 +5 + 5 + 8 + 8 ) = 268 2nd row : 50 8 + ( 2 + 2 + 3 + 3 + 4 + 4 + 8 + 9 ) = 435 3rd row : 60 5 + (0 + 5 +8 + 8 + 8 ) = 329 4th row : 70 5 + (0 + 1 + 3 + 5 + 8 ) = 367 5th row : 80 7 + (0 + 1 + 1 + 3 + 4 + 5 + 7 ) = 581 6th row : 90 5 + (1 + 3 + 4 + 6 + 9 ) = 473
Mean 268+ 435 + 329 + 367 + 581 + 473 = 36 2453 = 36 = 68.1(3sigfig )
Mode = 68
(d) Median is the average of the 18th and 19th score. Arrange the score of 3rd row : 0, 5, 8, 8, 8 15th, 16th, 17th, 18th, 19th Median = 68
Skill Practice 13A ( Part II) Refer to text book Page 136 Do the following questions in Exercise Book A Q2, 5, 6, 8, 12 and 13
Supplementary Exercise
1. Determine the mean, median and mode of the following table.
x 0 1 2 3 4 5 6 f 1 2 2 3 4 5 4
2. The following table shows the monthly wages of 27 employees in a certain factory in 1991.
Wages( $x) 670 760 850 960 1000 1200 No of employees (f) 4 9 8 3 2 1
Find (a) the mean monthly wage, (b) the median monthly wage, (c) the modal monthly wage.
3. The table below shows the frequency distribution of the number of spelling mistakes in a composition made by each pupil in a class of 36.
No of mistakes (x) 0 1 2 3 4 5 6 7 No of pupils (f) 3 7 10 6 5 3 1 1
Find (a) the mean, (b) the median, (c) the mode of the distribution.
4. A six-sided dice is thrown 29 times. The results are shown in the table below.
Number shown on dice 1 2 3 4 5 6 Frequency 8 7 5 2 3 4
For these results, write down (i) the mode, (ii) the median, (iii) the mean 5. The histogram shows the distribution of 200 numbers from 1 to 6 inclusive.
70 60
y 50 c
n 40 e u
q 30 e r 20 F 10 0 1 2 3 4 5 6 Num ber
From the distribution, find (a) the mode, (b) the median (c) the mean.
Comparison of the Mean, Median and Mode
The arithmetic mean is the most widely used measure of central tendency. It is usually preferred over the median and the mode. It is the most reliable measure provided there are no extreme values in the data because all the values in the data are used in calculating the mean unlike the mode and the median. Whenever the set of data contains extreme values, the median or mode will probably be a better indicator of the whole set of data because they are not influenced by extreme values.
The median may be the preferred measure of central tendency for describing economic, sociological and educational data. It is popular in the study of the social sciences because much of the data in the social sciences contain extreme values.
The mode is more useful in business planning as a measure of popularity that reflects central tendency or opinion.