ADVANCED MATHEMATICS FORM 6 – STATISTICS 1

Is the branch of Mathematics which deals with the collection, presentation and analysis of data obtained from various experiments?

FREQUENCY DISTRIBUTION

Is the arranged data summarized by distributing it into classes or categories with their frequency

Example;

GRAPHICAL REPRESENTATION

Is often useful to represent frequency nglish-swahili/distribution” target=”_blank”>distribution by means of diagrams.There are different types of diagrams. These are;

1. Line graph

2. Cumulative frequency curve (Ogive)

3. Circles or Pie

4. Bar chart (Histogram)

5. Frequency polygon

TYPES OF DATA

They are;

i)      Ungrouped.

ii)    Grouped data.

UNGROUPED DATA

Is the type of data in which each value is taken separately which represent to each other e.g. 20, 30 ,40 ,50  etc

REPRESENTATION OF UNGROUPED DATA

-Ungrouped data can be represented by;

(a) Frequency nglish-swahili/distribution” target=”_blank”>distribution table

(b) Cumulative frequency nglish-swahili/distribution” target=”_blank”>distribution table

(c) Frequency histogram

(d) Frequency polygon

(e) Frequency curve

(f) Cumulative frequency nglish-swahili/distribution” target=”_blank”>distribution curve (O gives)

A.FREQUENCY DISTRIBUTION TABLE

Is the table of values with their corresponding frequencies

For instance;

B. CUMULATIVE FREQUENCY TABLE

  Is the table of values with their corresponding cumulative frequencies

For instance;

C. FREQUENCY HISTOGRAM

Is the graph which is drawn by using frequency against given values.

Example

 

D.FREQUENCY POLYGON

Is the polygon which is drawn by using the corresponding points of frequencies against a given value.

Example

E. THE FREQUENCY CURVE

Is the curve which is drawn by joining (free hand) the corresponding points of frequency against given values.

Example

F.CUMULATIVE FREQUENCY CURVE (O GIVE)

Is the curve of cumulative frequency against given values

Example

MEASURES OF CENTRAL TENDENCY

These are;

I)    Mean

ii)     Mode

iii)   Median

     iv)   Harmonic mean

    v)   Geometric mean

1. MEAN

Is obtained by adding together all the items and dividing by the number of items.

Mean  () =

2. MODE

Is the number (value) which occur most frequently

For instance

i) Given  2,3,5,5,6,7,7,7,9

→7 has a frequency of 3

→Hence 7 is the mode

ii) Given 1,2,3,4,5,5,6,7,

→then 4 and 5 are the mode their frequency is 2

iii) Given 2, 33, 4, 5, 6, 7= there is no mode at all

3. MEDIAN

Is the middle number (value) when the data is arranged in the order of site.

N.B

I)  When the total number of items is ODD say “N” the value of

      Items give the mode.

II) When the total number of items is EVEN, say “N” there are two middle items, then the mean of the values of ½ N^th and (½ N+1)^th item is the median

4. HARMONIC MEAN

Is the reciprocal of arithmetic mean of their values.

– If it is harmonic mean

Then 

For value of x₁, x₂ ,x₃…x_n with the frequency, f1,f2, f₃+…….. f_n respectively

The harmonic mean is given by

          

5. GEOMETRIC MEAN

For the values of x₁, x₂, x₃, x₄,………x_n then the geometric mean is given by

Examples

Find the mean of 20, 22, 25, 28, and 30

Solution

Given 20, 22, 25, 28, 30

∑f =5

Mean () =

() =25

2. Find the mean of the following;

Solution

Consider the table below;

                                                ∑f=25                                       ∑ f(x)= 320

Mean () =

3. Find the median of 6, 8, 9, 10, 11, 12 and 13.

Solution

Given 6, 8, 9, 10, 11, 12, 13

 N=7

Position = 

=

= 4^th

Median is 10

4. Find the mode of the following items

0,1,6,7,2,3, 7,6,6,2,6,0, 5, 6, 0

Solution

From the given data 6 has a frequency of 5

→6 is the mode

5. Find the geometric mean of 4,16,8

Solution

Number 4,8,16

N=3

From

    G= 8

→Geometric mean is 8

6. Calculate the harmonic mean of the data 4,8,16

Solution

Numbers4, 8, 16

N=3

         

H = 6.857

The harmonic mean =6.857

B.MEASURES OF DISPERSION (Variability)

These are;
i) Variance

ii)  Standard deviation

iii)    Mean deviation

1.   VARIANCE

This is given by;

The formula is used for ungrouped data

Also,

Var(x) =

2.  STANDARD DEVIATION

This is given by;

S.D =  

S.D=

S.D

3.  MEAN DEVIATION

This is given by

M.D =

Also f =1

M.D =

Examples

7. From the nglish-swahili/distribution” target=”_blank”>distribution 1, 2, 3, 4, 5 find variance

8. Given the nglish-swahili/distribution” target=”_blank”>distribution 2, 3, 4, 5, 6, 7, 4, 5, 3 find

a) Variance

b) Standard deviation

c) Mean deviation

Solution

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

Variance

Form

 

 

Var(x) = 2.223.

b) Standard deviation

S.D =

S.D =

S.D = 1.491

c). Mean deviation

=

= 1.25556