MEASURES OF CENTRAL TENDENCY

The arithmetic mean, also popularly referred to as the “mean” is the average of a series of figures or values. The arithmetic mean can also be prepared for grouped data. In this case, the class mark (mid-point) of the individual class interical is used for the X – column

Formula used is

Arithmetic Mean

Example

Calculate the mean of the following marks scored by students in an econo

20, 12, 18.

Use a class interval of 0-9, 10-19, 20-29, e.t.c.

Solution

Frequency table for marks scored by students in Economics Examination

Scores (grouping)	Class marks (X)	Tally	Frequency (F)	FX
0-9	4.5	III	3	13.5
10-19	14.5	IIII   IIII	9	130.5
20-29	24.5	IIII	6	147
30-39	34.5	IIII	5	172.5
40-49	44.5	IIII	4	178
50-59	54.5	II	2	109
60-69	64.5	I	1	64.5

The Median

The median is defined as an average, which is the middle value when figures are arranged in order of magnitude.

When the items are large, it may be necessary to use other methods other than arranging in order of magnitude to calculate the median. This will require that a frequency table be prepared.

Hence, from a frequency distribution, the median is calculated thus;

Member for even number of items i.e. N is even.  Where N is the summation of all the frequency and this is the terminal cumulative frequency.

Example 1

Use the information in the table: Calculate the median age of ssII students

Age Distribution of SSII Students

Age (yrs)	6	8	10	11	12	13	14
Frequency	5	10	3	8	7	10	8

Solutions

Cumulative frequency for age Distribution of SSII Students

Age Distribution of SSII Students

Age (yrs)	6	8	10	11	12	13	14
No of students(Frequency)	5	10	3	8	7	10	8

Example 2

The data in table represents the marks scored by Economics students in NECO examination.  Calculate the median score.

Marks scored by Economics students in

Marks %	12	18	24	30	36	40	48
Frequency	6	1	10	8	12	3	4

Solution

Cumulative frequency of table for Marks scored by Economics students in NECO Examination

Marks %	12	18	24	30	36	40	48
Frequency	6	1	10	18	12	3	4
Cumulative Frequency	6	7	17	25	37	40	44

From the table, there are 44 members as indicated by the terminal (last) cumulative frequency. Since this 44 is even, the median score will be;

Mode for Grouped Data

Example

The table below shows the distribution of the weight of students in a certain school.

Weight (kg)	40-44	45-49	50-54	55-59	60-64	65-69
Frequency	4	11	15	9	3	8

Obtain the modal of the weight.

Class Boundary          Frequency

39.5 – 44.5                           4

44.5 – 49.5                           11

49.5 – 54.5                            15

54.5 – 59.5                           9

59.5 – 64.5                           3

64.5 – 69.5                           8

From the table, the modal class has frequency of 15.  The class boundaries are 49.5 -54.5. Therefore, the lower boundary of the modal class is 15, while the frequency is before and after it are 11 and 9 respectively.

Questions

Marks (kg)	55-59	60-64	65-69	70-74	75-79	80-84	85-89	90-94	95–99	100-104
Frequency	2	6	9	23	25	13	10	6	5	1

1. State the modal class
2. Estimate the mode of the distribution, correct to one decimal place
3. The frequency table below represents the number of oranges picked by 20 students.

Calculate the median of the table.

X	0	1	2	3	4	5
F	2	5	6	4	2	1