MEASURES OF CENTRAL TENDENCY

WEEK ONE

MEASURES OF CENTRAL TENDENCY

CONTENT

• MEAN
• MODE
• MEDIAN

In mathematics, the mean, median, and mode are all ways of describing central tendency. The mean is the most common type of average, and is calculated by adding up all of the numbers in a set and dividing by the total number of items. The median is the middle number in a set of numbers, and the mode is the number that appears the most often.

Each of these measures of central tendency has its own strengths and weaknesses. The mean is the most sensitive to outliers, while the median is less sensitive to outliers but can be affected by symmetry in the data. The mode is the most robust to outliers, but can be affected by clustering in the data.

In practice, the mean is the most commonly used measure of central tendency. It is easy to calculate, and is not affected by the distribution of the data. However, it can be distorted by a few extreme values. The median is a good choice when there are a few extreme values, or when the data is not symmetrical. The mode is useful when there is clustering in the data, or when the data is not normally distributed.

MEASURES OF CENTRAL TENDENCY:  are the values which show the degree to which a given data or any given set of values will converge toward the central point of the data.

Measures of central tendency, also called measures of location, is the statistical information that gives the middle or centre or average of a set of data. Measures of central tendency include arithmetic mean, median and mode.

MEAN: This is the average of variables obtained in a study. It is the most common kind of average. For group data the formula for calculating the mean is ∑fx.

                                                            ∑f

Where, Ʃ =Summation

              F=frequency

              X=observation

MEDIAN: It is the middle number in any given distribution. The formula is

Median = L + (N\2-Fb)c

                               f   

Where; L = Lower class limit.

            N = Summation 0f the frequency.

           Fb = Cumulative frequency before the median class.

             f = frequency of the median class.

             c= Class size.

MODE: It is the number that appears most in any given distribution, i.e the number with the greatest frequency. When a series has more than one mode,say two,it is said to be bi-modal or tri-modal for three.

Mode= L  +    D1     

                     D1+D2           

Where, M=mode

              L=the lower class boundary of the modal class.

            D1=the frequency of the modal class minus the frequency of the class before the modal class.

            D2=the frequency of the modal class minus the frequency of the class after it.

              C=the width of the modal class.

Example: The table below shows the marks of students of JSS 3 mathematics.

Use the information above to calculate the following:                 

1. the mean
2. the median
3. the mode

Solution

mark    frequency    mid-point    fx

                                            27                                                                                        506                                                                                                                                                                         

1. A. Mean= ∑fx = 506\27 

      Ʃf =18.7

1. B. median

L1= 15.5

N\2 =27\2=13.5

Fb =9

F =5   

C= 5

M=15.5+ (13.5-9)5

          5

M=20

1. C. mode= L+     D1

                          D1+D2

L1=20.5

D1=7-6=1

D2=7-0=7

C=5

M=25.5+ (1\1+7)5

M=26.125.

EVALUATION

The table below shows the weekly profit in naira from a mini-market.

You are required to calculate:

1. The mean.
2. The median.
3. The mode.

READING ASSIGNMENT

1. Amplified and Simplified Economics for SSS by Femi Alonge page 29-30.
2. Further Mathematics Scholastics Series page 265-265.

GENERAL EVALUATION QUESTIONS

1. Outline the merits of a Joint Stock Company
2. The number of observations smaller than ________ _______ divides the data into four equal parts.
3. 1. 1. 1. Median
         2. Quartiles
         3. Mean
         4. None of the above

   Answer: b

   1. 1. What is the mean of the following numbers: 23, 45, 87, 40, 50?
         1. 49
         2. 34
         3. 56
         4. None of the above

   Answer: a

   1. 1. Which of the following is a characteristic of a mean?
         1. The sum of deviations from the mean is zero
         2. It minimises the sum of squared deviations
         3. It is affected by extreme scores
         4. All of the above

   Answer: d

   1. 1. Percentiles divide a series into ______.
         1. Ten equal parts
         2. Twenty equal parts
         3. Fifty equal parts
         4. Hundred equal parts

   Answer: d

   1. 1. Which of the following diagrams is used to find the value of mode graphically?
         1. Pie chart
         2. Bar graph
         3. Histogram
         4. None of the above

   Answer: c

4. Describe the problems facing Agriculture in Nigeria.
5. Outline the main features of Malthusian theory of Population.
6. What is money?   
7. List five characteristics of money.

WEEKEND ASSIGNMENT

1. Which of the following is not a set of measure of central tendency? (a) mode and mean (b) mean and median (c) mean and percentile (d) mode and median
2. The  most frequently occurring value in a give data is (a) mean ( b ) mode (c ) range (d) median.
3. The formula (n+1)th is for calculating (a ) median (b ) mode (c ) mean (d) range.

        2

1. The L1 in the formula for the calculation of measures of location represents……… (a)   lower class boundary of the median class (b) actual frequency of the modal class (c) upper class boundary of the median class (d) frequency of the class just after the median class 
2. The formation of cumulative frequency is necessary for the calculation of………… (a)  mean (b) range (c) median (d) mode

SECTION B

The following table shows the distribution of marks scored by a class of students in a promotion examination.

1. Calculate the mean mark.
2. Find the mode.