Statistics SS2 Mathematics Lesson Note

MEASURES OF CENTRAL TENDENCY:  

These 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, are the statistical information that gives the middle centre or average of a set of data. Measures of central tendency include arithmetic mean, median and mode.

1. 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

2. 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.

3. 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

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:

Total:  F= 27                           fx=  506

1. Mean= ∑fx = 506 ÷ 27 

                   Ʃf =18.7

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. 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.

MEASURES OF DISPERSION:  

They are also known as measures of spread or variation that describe how the data given in any distribution are spread about the Mean or the overall spread of the data. These measures are the range, mean deviation, standard deviation, variance, coefficient of variation, etc.

1. The Range: The range of data is the difference between the highest and the lowest value in the data. The formula for the calculation of the range is:

Range = Highest value – Lowest value

2. The Mean Deviation: This measures the dispersions around the arithmetic mean. It tells us how far, on average, the individual observations are from the mean.  For a grouped frequency distribution 

           Mean Deviation =✓∑f/x- x/  

                                ✓∑f               

3. Variance: This is the average of the square about the deviations of the measurement about their mean.

Variance = ∑f(X – X)2

            ∑f                 

4. Standard Deviation: This is the square root of the average of the squares of the deviations of the measurement about their mean.    

Standard Deviation = ∑f(X – X )2

                                          Ʃf             

Example: 

The data below shows the weight of 50 students to the nearest kg.

65 58 51 36 23 40 53 59 70 51 46 59 50 67 46 39 61 62 73 60 71 51 47 32 48 40 40 51 58 67 60 69 43 52 37 26 38 50 59 40 44 54 42 68 74 45 39 48 55.

Prepare a grouped frequency table 

Calculate: 

1. The range 
2. The mean deviation 

iii. The variance 

1. The standard deviation 

NB: Note that the standard Deviation is the positive square root of the variance.

∑fx = 2535 = 50.7

∑f        50

Range = Highest score –Lowest score 

= 74 – 23 = 51 kg

 Mean Deviation 

=∑f/ X –X/ = 529.60 = 10.59kg

    ∑f                 50 

(iii) Variance = ∑f/X – X /2

                                ∑f               

= 7848 = 156.9kg

     50                              

(iv) Standard Deviation            

✓∑f/X – X /2                                 

✓∑f

✓156.96

✓50                                                    =12.53kg