Worked examples and exercises are in the text STROUD PROGRAMME 27 STATISTICS

Worked examples and exercises are in the textSTROUD

PROGRAMME 27

STATISTICS


Programme 27: Statistics

Introduction

Arrangement of data

Histograms

Measure of central tendency

Dispersion

Frequency polygons

Frequency curves

Normal distribution curve

Standardized normal curve



Introduction

Arrangement of data

Histograms


Dispersion

Frequency polygons

Frequency curves





Introduction

Statistics is concerned with the collection, ordering and analysis of data.

Data consists of sets of recorded observations or values. Any quantity that can have a number of values is a variable. A variable may be one of two kinds:

(a) Discrete – a variable whose possible values can be counted

(b) Continuous – a variable whose values can be measured on a continuous scale



Introduction

Arrangement of data

Histograms


Dispersion

Frequency polygons

Frequency curves





Arrangement of data

Table of values

Tally diagram

Grouped data

Grouping with continuous data

Relative frequency

Rounding off data

Class boundaries



Arrangement of data

Table of values

A set of data:

Can be arranged in ascending order:

28 31 29 27 30 29 29 26 30 28

28 29 27 26 32 28 32 31 25 30

27 30 29 30 28 29 31 27 28 28

25 26 26 27 27 27 27 28 28 28

28 28 28 28 29 29 29 29 29 29

30 30 30 30 30 31 31 31 32 32



Arrangement of data

Table of values

Once the data is in ascending order:

It can be entered into a table.

The number of occasions on which anyparticular value occurs is called the frequency, denoted by f.

25 26 26 27 27 27 27 28 28 28

28 28 28 28 29 29 29 29 29 29

30 30 30 30 30 31 31 31 32 32 Value Number of times

25 1

26 2

27 4

28 7

29 6

30 5

31 3

32 2



Arrangement of data

Tally diagram

When dealing with large numbers of readings, instead of writing all the values in ascending order, it is more convenient to compile a tally diagram, recording the range of values of the variable and adding a stroke for each occurrence of that reading:



Arrangement of data

Grouped data

If the range of values of the variable is large, it is often helpful to consider these values arranged in regular groups or classes.



Arrangement of data

Grouping with continuous data

With continuous data the groups boundaries are given to the same number of significant figures or decimal places as the data:



Arrangement of data

Relative frequency

If the frequency of any one group is divided by the sum of the frequencies the ratio is called the relative frequency of that group. Relative frequencies can be expressed as percentages:



Arrangement of data

Rounding off data

If the value 21.7 is expressed to two significant figures, the result is rounded up to 22. similarly, 21.4 is rounded down to 21.

To maintain consistency of group boundaries, middle values will always be rounded up. So that 21.5 is rounded up to 22 and 42.5 is rounded up to 43.

Therefore, when a result is quoted to two significant figures as 37 on a continuous scale this includes all possible values between:

36.50000… and 37.49999…



Arrangement of data

Class boundaries

A class or group boundary lies midway between the data values. For example, for data in the class or group labelled:

7.1 – 7.3

(a) The class values 7. 1 and 7.3 are the lower and upper limits of the class and their difference gives the class width.

(b) The class boundaries are 0.05 below the lower class limit and 0.05 above the upper class limit

(c) The class interval is the difference between the upper and lower class boundaries.



Arrangement of data

Class boundaries

(d) The central value (or mid-value) of the class interval is one half of the difference between the upper and lower class boundaries.



Introduction

Arrangement of data

Histograms


Dispersion

Frequency polygons

Frequency curves





Histograms

Frequency histogram

Relative frequency histogram



Histograms

Frequency histogram

A histogram is a graphical representation of a frequency distribution in which vertical rectangular blocks are drawn so that:

(a) the centre of the base indicates the central value of the class and

(b) the area of the rectangle represents the class frequency



Histograms

Frequency histogram

For example, the measurement of the lengths of 50 brass rods gave the following frequency distribution:



Histograms

Frequency histogram

This gives rise to the histogram:



Histograms

Relative frequency histogram

A relative frequency histogram is identical in shape to the frequency histogram but differs in that the vertical axis measures relative frequency.



Introduction

Arrangement of data

Histograms


Dispersion

Frequency polygons

Frequency curves






Mean

Mode of a set of data

Mode of a grouped frequency distribution

Median of a set of data

Median with grouped data



Mean

The arithmetic mean:

of a set of n observations is their average:

When calculating from a frequency distribution, this becomes:

x

sum of observationsmean = that is

number of observations

xx

n

xf xfx

n f



Mode of a set of data

The mode of a set of data is that value of the variable that occurs most often.

The mode of:2, 2, 6, 7, 7, 7, 10, 13

is clearly 7. The mode may not be unique, for instance the modes of:

23, 25, 25, 25, 27, 27, 28, 28, 28

are 25 and 28.




The modal class of grouped data is the class with the greatest population.

For example, the modal class of:

Is the third class.




Plotting the histogram of the data enables the mode to be found:




The mode can also be calculated algebraically:

If L = lower boundary value l = AB = difference in frequency on the lower boundary u = CD = difference in frequency on the upper boundary c = class interval

the mode is then:

mode l

L cl u





L = ...... l = .....u = ...... c = .....

mode l

L cl u





L = 15.5 l = 16-7 = 9u = 3 c = 16-10=6

mode l

L cl u

15.5+9/(9+6)*3 = 15.5 + 1.8 = 17.3



Median of a set of data

The median is the value of the middle datum when the data is arranged in ascending or descending order.

If there is an even number of values the median is the average of the two middle data.



Median with grouped data

In the case of grouped data the median divides the population of the largest block of the histogram into two parts:

In this frequency distribution A + B = 20so that A = 7:

Therefore, Median = 30.85 + 0.105 = 30.96

6 12 15 13 9 5A B

7The width of class interval

200.35 0.3

0.105

A

A B



Introduction

Arrangement of data

Histograms


Dispersion

Frequency polygons

Frequency curves





Dispersion

Range

Standard deviation

Alternative formula for the standard deviation



Dispersion

Range

The mean, mode and median give important information about the central tendency of data but they do not tell anything about the spread or dispersion about the centre.

For example, the two sets of data:

26, 27, 28 ,29 30 and 5, 19, 20, 36, 60

both have a mean of 28 but one is clearly more tightly arranged about the mean than the other. The simplest measure of dispersion is the range – the difference between the highest and the lowest values.



Dispersion

Standard deviation

The standard deviation is the most widely used measure of dispersion.

The variance of a set of data is the average of the square of the difference in value of a datum from the mean:

This has the disadvantage of being measured in the square of the units of the data. The standard deviation is the square root of the variance:

2 2 21 2( ) ( ) ( )

variance nx x x x x x

n

2

1

( )standard deviation

n

ii

x x

n



Dispersion

Alternative formula for the standard deviation

Since:

That is:

2 2 2

1 1

2 2 2 2 2

1 1 1 1

2

21

( ) ( 2 )

2 2

n n

i i ii i

n n n n

i i ii i i i

n

ii

x x x x x x

n n

x x x x x nx nx

n n

xx

n

2 2x x

Documents

Worked examples and exercises are in the text STROUD PROGRAMME 27 STATISTICS