ADVANCED MATHEMATICS FORM 6 – STATISTICS 1

MEASURES OF GROUPED DATA

These are;

a) Measures of central tendency

b) Measures of dispersion (variability)

c) Measures of position

A.MEASURES OF CENTRAL TENDENCY

These are i) mean

                ii) Median

                iii) Mode

I. MEAN ()

-By direct method

Mean ( )

Where

∑f(x) → is the sum of frequencies times class mark

∑f →sum of frequency

By assumed mean method

I.e. mean(x) =

Where

X → is the class mark

A → is the assumed mean

D → deviation

 Substitute (ii) into (i)

BY CODING METHOD

Where; 

             d →  is the deviation

           C →   is the class size

           n →  is the coding number

Substitute (ii) into (i)

Examples

1.     From the following nglish-swahili/distribution” target=”_blank”>distribution

Find the mean by

i) Direct method

ii) assumed mean

iii) Coding method

Solution

Consider the nglish-swahili/distribution” target=”_blank”>distribution table

Let A = Assumed mean

            = 30

C =Class size

     = 10

a) By direct mean method

I.e. Mean ( )

Mean (x) = 25.30

b) By assumed mean method

c) By coding method

MODE

This is the value which occurs most frequently in grouped data mode can be determined by using two methods

a) By estimation from histogram

b) By calculation method

A) MODE FROM HISTOGRAM

-Consider the three bars under consideration of the highest bar with their two adjacent bars from the histogram.

                                                                                                             

BY CALCULATION METHOD

-Assume that the figure below represents three rectangles of the histogram of the frequency nglish-swahili/distribution” target=”_blank”>distribution of central rectangle corresponding to modal class.

Where

D₁ → is the difference between the frequencies of the mode class and the frequency of the class just before the modal class.

D₂ → is the difference between the frequency of the modal class and the frequency of the class just after the modal class

L_c → is the lower class boundary of the modal class

U → is the upper class boundary of the modal class

Hence from the histogram

D₂(X – L_C) = D₁ (U – X)

D₂ X – D₂L_C =D₁U – D₁X

D₁X + D₂X = D₂L_C + D₁U

X (D₁ + D₂) = D₂ L_C + D₁U

But

U L_C = class size(C)

C= U L_C

U = C + L_C

X(D₁ + D₂ ) = D₂L_C + D₁ (L_C+ C)

2. MEDIAN

Position of the median class N/2 hence therefore using interpolation

Where;

L_C→ is lower boundary of the median class

→ Sum of the frequency before the median class

fm→ frequency of the modal class

c → class size

Example

1.     from the following nglish-swahili/distribution” target=”_blank”>distribution table

 Find i) mode

        ii) Median

2.     Find the mean and the median o the following nglish-swahili/distribution” target=”_blank”>distribution

3.     Given the frequency nglish-swahili/distribution” target=”_blank”>distribution table below

Solution 1

Consider the nglish-swahili/distribution” target=”_blank”>distribution table

i)   mode from the table

Modal class = 15→21

ii)   median

Position of median class

=

=40

Median class =15→21

L_C= (15 – 0.5)

= 14.5

∑fb=18

Fm=22

C=7

Then

Median = 21.5

MEASURES OF DISPERSION (variability)

These are

       i) Variance

ii)  standard deviation

VARIANCE

→Variance by direct mean method

Recall

VARIANCE BY ASSUMED MEAN METHOD

Recall

Put X – A= d

X = A + d…… (ii)

Where

       X → class mark

  A → Assumed mean

Substitute……..ii)………..i) as follows

       

VARIANCE BY CODING METHOD

Recall,

Where;

                D = deviation

              C = class size

              U = coding number

Substitute (ii) into (i)

2.  STANDARD DEVIATION

This is given by  

→By direct mean method

→By assumed mean method

→By Coding method

questions

1.  Given the nglish-swahili/distribution” target=”_blank”>distribution class interval frequency

Find the standard deviation by coding method.

2.  The table below shows the frequency nglish-swahili/distribution” target=”_blank”>distribution of intelligence quotient (IQ)of 500 individuals

Using coding method find

i)  Mean

ii) Standard deviation

c.      MEASURES OF POSITION

These are
1.quartile

   2. deciles

  3. Percentile

1.QUARTILE

Recall

2.   DECILE

Recall

3.  PERCENTILE

Recall

Note
2^nd quartile 5^th, decile and 50^th percentile are MEDIANS