Application Of Measures Of Central Tendency To Analyse Given Information JSS3 Mathematics Lesson Note

CALCULATION OF RANGE, MEAN, MEDIAN, AND MODE OF UNGROUPED DATA

RANGE

The range of a set of numbers is the difference between the largest and the smallest numbers.

Example: Find the range of the following set of scores: 79, 50, 52, 34, 68, 84.

Solution: Arrange the data in rank order: 79, 68, 60, 52, 50, 34.
The range is 79 – 34 = 45.

THE MEAN

There are many kinds of averages. The mean or arithmetic mean is the most common kind. If there are n numbers in a set, then:

Mean=∑of the numbers in the setn\text{Mean} = \frac{\sum \text{of the numbers in the set}}{n}Mean=n∑of the numbers in the set​

Examples:

1. Calculate the mean of the following set of numbers

176, 174, 178, 181, 174
175, 179, 107, 177, 102

Solution:

Mean= 176+174+178+181+174+175+179+107+177+102 =1723=177.6

                                                10                                            10

1. Five children have an average age of 7 years 11 months. If the youngest child is not included, the average increases to 8 years and 4 months. Find the age of the youngest child.

Solution:

Total age of all five children:
= 5 x 7 y 11 mo
= 35 y 55 mo
= 39 y 7 mo
= 39 y 7 mo

Total age of the four older children:
= 4 x 8 y 4 mo
= 32 y 16 mo
= 33 y 4 mo

Age of the youngest child:
= 39 y 7 mo – 33 y 4 mo
= 6 y 3 mo

EVALUATION:

1. Find the mean of the numbers: 13, 2, 8, 5, and 11. Also, find the range of the set of numbers.
2. A mother has seven children. The mean age of the children is 13 years 2 months. If the mother’s age is included, the mean age rises to 17 years 7 months. Calculate the age of the mother.

MEDIAN AND MODE

MEDIAN:

If a set of numbers is arranged in order of size, the middle term is called the median. If there is an even number of terms, the median is the arithmetic mean of the two middle terms.

Examples:

Find the median of:
a) 15, 11, 8, 21, 17
b) 38, 21, 14, 4, 43, 9, 2, 50

Solution:

8. a) Arrange the numbers in rank order (i.e. from highest to lowest):
   21, 17, 15, 11, 8.
   There are seven numbers. The median is the 4th number, 11.
9. b) Arrange the numbers from the lowest to highest:
   2, 9, 14, 21, 38, 43, 50.
   There are six numbers. The median is the mean of the 3rd and 4th terms.

Median=21+382=47\text{Median} = \frac{21 + 38}{2} = 47Median=221+38​=47

MODE: The mode is the number or value that appears most often, i.e. the number with the greatest frequency.

Example: Twenty-one students experimented to find the melting point of naphthalene. The table below shows the results. What was:

1. a) the modal temperature?
   b) the median temperature?

| Temperature (°C) | 78 | 79 | 80 | 81 | 82 | 83 | 60 |
| Frequency | 1 | 2 | 7 | 5 | 3 | 2 | 1 |

Solution:

1. a) Seven students recorded a temperature of 80°C. This was the most frequent result.
   Mode = 80°C.
2. b) There were 21 students. The median is the 11th. If the temperatures were written down in order, there would be one of 79°C, two of 79°C, seven of 80°C, and so on. Since 1 + 2 + 7 = 10, the 11th temperature is one of the five 81°C.
   Median = 81°C.

EVALUATION

For the following set of numbers:
12, 14, 14, 15, 18, 18, 19, 19, 21

1. a) State the median.
   b) State the mode.
   c) Calculate the mean.

WEEKEND ASSIGNMENT

The number of goals scored by a team in nine handball matches are as follows: 3, 5, 7, 7, 8, 4, 1, 1, 5.

Which of the following statements are true of these scores?

1. a) The mean is greater than the mode.
   b) The mode and the median are equal.
   c) The mean, median, and mode are all equal.