STATISTICS

By the end of the subtopic learners should be able to:
collect statistical data.tabulate statistical data. represent statistical data on bar charts, pictograms, pie charts and so on.make simple inferences using the graphs and chart. calculate the measures of central tendency from given data.

Statistics

Statistics is the collection, analysis, interpretation and representation of numerical data.

Data collection

Raw data can be collected by asking questions or observation.
Examples of data collection

1. Shoe sizes in a Form 1A class.
2. Colours of cars passing at a certain bus stop.
3. Means of transport used by pupils to school.

Example of raw data

The following data was collected from 20 pupils in Form 1A class of their shoe sizes.

The data is not arranged in any order and we may not be able to quickly tell the number of times a shoe size appears. There is no pattern in the data.

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Frequency table

The data above can be put together in a table as follows:

4       5       4       7       3       5       4       6       7       6       5       5         5       5       4       4       7       5       4       5

Shoe size	3	4	5	6	7
No. of students (Frequency)	1	6	8	2	3

Frequency shows how many times data appears. The total frequency is the total number of items (pupils) in the data collected.

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Statistical Data Presentation

Information can be presented in different ways:

Graphs and charts

Mean median and mode

The mean

• Mean = $\frac{\mathrm{The} \mathrm{sum} \mathrm{of} \mathrm{the} \mathrm{numbers} \mathrm{in} \mathrm{a} \mathrm{set}}{\mathrm{n}}$ where  is the number of members in a set.
• The mean is the average of a given set of numbers.
• It may not be a value from the original list of numbers.

The median

• The middle term in an ordered set of values.
• The order should be either ascending order or descending order.
• If the total elements in an ordered set are even :
$\frac{{\displaystyle \underset{}{∑}\left(\mathrm{middle} \mathrm{terms}\right)}}{2}$

The mode

• The mode is the number that appears most times than others in the set. In other words, it is the number with the highest frequency.

Example

Find the mean, mode and the median of the following sets of numbers giving your answers correct to 1d.p:

• 22; 15; 17; 15; 19; 22; 22; 13; 21; 20; 14
• 2,4;  3,3;  1,1;  2,2;  1,1;   1,1;  1,8; 2,0

Answers:

a.)

i.) Mean = $\frac{22+15+17+15+19+22+22+13+21+20+14}{11}$

= $\frac{200}{11}$

= 18,2

ii.) Mode = 22

iii.) To find median first arrange the numbers in order

13; 14; 15; 15; 17; 19; 20; 21; 22; 22; 22

Median = 19

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b.)

i.) Mean = $\frac{2,4+3,3+1,1+2,2+1,1+1,1+1,8+2,0}{9}$

$=\frac{15}{8}$

= 1,9

ii.) Mode = 1,1

iii.) Arranging the number in order 1,1; 1,1; 1,1; 1,8; 2,0; 2,2; 2,4; 3,3

Take the middle pair in this case, add them and divide by 2

Median =$\frac{1,8+2,0}{2}$

= 1,9

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Assumed mean

• Assumed mean is the estimated or guessed mean.
• Deviation is the difference between the assumed mean and the numbers in the set, it can be positive or negative.

Calculating mean using the assumed mean:

Example:

Find the mean of the following numbers by means of assumed mean:
2,1;  2,3;  2,0;  2,7;  2,6;  2,2;  2,5;  2,4;  2,8

Answer:

Assumed mean =2,5

Calculating the deviation from the assumed mean:

	Deviation
	-	+
2,1 — 2,5	0,4
2,3 — 2,5	0,2
2,0 — 2,5	0,5
2,7 — 2,5		0,2
2,6 — 2,5		0,1
2,2 — 2,5	0,3
2,5 — 2,5	-	-
2,4 — 2,5	0,1
2,8 — 2,5		0,3
Sum of deviations	1,5	0,6
	0,9
Sum of deviations = -0,9

Mean = 2,5 + $\frac{\left(-0,9\right)}{9}$

= 2,5 — 0,1

= 2,4