Lesson 3: Basic Tools of Economic Analysis

Introduction
Generally, the basic tool adopted ¡n economic analysis ¡s statistics. Thus, data generated overtime or in respect of empirically examined economic problem is analysed by using statistical tools as the instrument of analysis, in this regard, statistics is central to economic analysis.

Some of these basic tools used for economic analysis are tables, graphs, charts, mode, median, mean, range and standard deviation.

Unit 1 Table
A table is defined as a systematic and orderly arrangement of information, facts or data using rows and columns for presentation, which makes it easier for better understanding. Tables serve as the most commonly used tools for economic analysis.

Let us consider the table in the figure below:

Class Boys Girls
JS1 15 18
JS2 14 17
JS3 10 14
SS1 11 10
SS2 10 8

It represents the population of boys and girls in JS1 to SS2 for a particular school arranged in rows and columns

Characteristics of a table
(i) The sub-headings for the columns and rows used must be stated
(ii) It must have a title or heading.
(iii) A table must be very simple.
(iv) The units of measurement used in the table must be stated.

Importance of tables
(i) A table eases comparison between different classes of data
(ii) It assists in an orderly arrangement of data
(iii) A table helps to summarise the data presented
(iv) It makes it easier and faster to make decision through the use of table

Unit 2: Graph
A graph is a visual representation of numerical information. Graphs condense detailed numerical information to make it easier to see patterns (such as “trends”) among data.  For example, which countries have larger or smaller populations?

A careful reader could examine a long list of numbers representing the populations of many countries, but with more than two hundred nations in the world, searching through such a list would take concentration and time. Putting these same numbers on a graph, listing them from highest to lowest, would reveal population patterns much more readily.

Economists use graphs not only as a compact and readable presentation of data, but also for visually representing relationships and connections—in other words, they function as models. As such, they can be used to answer questions.

Examples of graphs are line graphs, pie graphs (or charts) bar graphs, pictographs (or pictograms)

Features of Graph
(i) The unit of measurement must be indicated in a good graph
(ii) All graph should have X-axis on the horizontal side and Y-axis on the vertical side.
(iii) The X and Y axis of the graph must carry different variables
(iv) Graph must possess appropriate scales

Importance of Graph
(i) Graph makes clearer information presented in tabular form
(ii) Graph helps to interpret values of variables.
(iii) Graph makes it possible for changes in variable on quantities to be expressed.

Line Graph
A line graph, also known as a line chart, is a type of chart used to visualize the value of something over time. For example, the population of a school in each class in a week can be summarised using a line graph.

The line graph consists of a horizontal x-axis and a vertical y-axis. Most line graphs only deal with positive number values, so these axes typically intersect near the bottom of the y-axis and the left end of the x-axis. The point at which the axes intersect is always (0, 0).

Each axis is labeled with a data type. For example, the x-axis could be days, weeks, quarters, or years, while the y-axis shows revenue in dollars. Data points are plotted and connected by a line in a “dot-to-dot” fashion.

Example:
The data in the table below represents the population of a school in a week. Represent the information by a line graph

Day of the week Population
Monday 750
Tuesday 830
Wednesday 800
Thursday 700
Friday 650

Pie Charts (or Graphs)
A pie chart shows how a total amount is divided between levels of a categorical variable as a circle divided into radial slices. Each categorical value corresponds with a single slice of the circle, and the size of each slice (both in area and arc length) indicates what proportion of the whole each category level takes.

The pie chart is usually measure either in percentage or in degree with the aid of a mathematical device called a protractor. The entire circle is represented by 360o or 100% and each sector is measured in degree with the aid of a protractor.

Method of construction of Pie Charts
Step 1:
Add up the total figure of the product or value under consideration
Step 2: Work out the percentage or degree of the total which each component part represents.
Step 3: Draw a circle of a convenient size.
Step 4: Divide the circle up into sectors subtended by these angles calculated in step (2) above using a protractor
Step 5: Write the percentage or degree of each sector by it
Step 6: Use key or highlight where information cannot be written inside the circle to show what each sector represents.

Example
The data in the table below represents the population of a school in a week. Represent the information in a pie chart

Day of the week Population
Monday 750
Tuesday 830
Wednesday 800
Thursday 700
Friday 650

Solution:
Step 1: Add up the total values of all the population, i.e. 750 + 830 + 800 + 700 + 650 = 3,730
Step 2: Work out the percentage or degree of the total which each component part represents as shown in the table below:

Day of the week Population Working in Percentage (%) Workings in degree (0o) Angle of sector
Monday 750 \frac{750}{3730} x \frac{100}{1}  = 20.1% \frac{750}{3730} x \frac{360}{1} = 72.4o
Tuesday 830 \frac{830}{3730} x \frac{100}{1}  = 22.3% \frac{830}{3730} x \frac{360}{1}  = 80.1o
Wednesday 800 \frac{800}{3730} x \frac{100}{1} = 21.4% \frac{800}{3730} x \frac{360}{1} = 77.2o
Thursday 700 \frac{700}{3730} x \frac{100}{1} = 18.8% \frac{700}{3730} x \frac{360}{1} = 67.6o
Friday 650 \frac{650}{3730} x \frac{100}{1} = 17.4% \frac{650}{3730} x \frac{360}{1} = 62.7o
Total 3,730                      100%                       360o

Using the values in degree i.e. 72.4o, 80.1o, 77.2o, 67.6o, 62.7o construct the pie chart using a protractor as shown below:

Exercise

If the budget of the country was $ 7,200.00, how much is allocated to Education? A. $ 2,400.00   B. $ 2,000.00   C. $ 1,200.00     D. $ 1,000.00  [WAEC 2016]

Bar Charts (or Graphs)
Meaning: Bar chart or graph is a graph made up of bars of rectangles which are of equal width and whose lengths are proportional to the quantities they represent. The major characteristics of the bar chart are that the body of the bars must not touch each other. There must be a space or gap between one bar and another. Bar chart may be arranged vertically or horizontally.

Types of bar charts
There are three major types of bar charts. These are:
(i) Simple bar chart
(ii) Component bar chart and
(iii) Multiple bar chart.

(i) Simple bar charts; Simple bar chart is used when the data given are made up of only one item or component. The bar chart can be presented by tabulated data with evenly spaced bars, separated by gaps with the length proportional to the magnitude of the value given.

Example
The data in the table below represents the population of a school in a week. Represent the information in a bar chart

Day of the week Population
Monday 750
Tuesday 830
Wednesday 800
Thursday 700
Friday 650

Using a suitable scale, the graph can be plotted as shown below:

(ii) Component Bar Chart
A component bar chart is used when the data involved are of two variables.

Example:
The data in the table below represents the population of a school in a week. Represent the information in a bar chart

Days of the week Boys Girls Total
Monday 500 250 750
Tuesday 600 230 830
Wednesday 700 200 900
Thursday 400 300 700
Friday 400 250 650

Solution:
Using a suitable scale, the graph is drawn as shown below:

(iii) Multiple Bar Charts:
The multiple bar chart is used when there are about three or more variables in a given data, where each of the bar stands for a component variable.

Example:
The data in the table below represents the population of a school in a week. Represent the information in a bar chart

Day of the week Morning Afternoon Evening
Monday 500 250 150
Tuesday 600 230 130
Wednesday 700 200 100
Thursday 400 300 200
Friday 400 250 100

Using a suitable scale, the graph can be drawn as shown below:

Pictogram
Pictograms (or pictographs) are charts in which pictures or drawings of objects are used to represent items in a given data.

Also known as “pictographs”, “icon charts”, “picture charts”, and “pictorial unit charts”, pictograms use a series of repeated icons to visualize simple data. The icons are arranged in a single line or a grid, with each icon representing a certain number of units (usually 1, 10, or 100).

The pictures used are to represent the magnitude of the variables.

Histogram
A frequency distribution shows how often each different value in a set of data occurs. A histogram is the most commonly used graph to show frequency distributions. It looks very much like a bar chart, but there are important differences between them.

The height of each rectangle represents the magnitude of the data lying within each class interval. The area of the rectangle are proportional to the class frequencies. No space between two bars, unlike the bar chart.

Example: Represent the data above graphically using a histogram

Day of the week Population
Monday 500
Tuesday 600
Wednesday 700
Thursday 400
Friday 300

Solution: – using a reasonable scale, your histogram should look like the figure below:

Frequency Distribution
In statistics, a frequency distribution is a list, table or graph that displays the frequency of various outcomes in a sample. Each entry in the table contains the frequency or count of the occurrences of values within a particular group or interval.

Example: Represent the following scores in a frequency table:
20, 10, 5, 15, 5, 5, 8, 10, 7, 20, 10, 8, 8, 7, 7, 15, 10

Solution:
Frequency table

Score Frequency
5 3
7 3
8 3
10 4
15 2
20 2
Total 17

 

Unit 3: Measure of Central Tendency
A measure of central tendency is that which is concerned with measuring for representation out of a set of numbers or that which is typical of a given distribution in such a set of numbers or data. There are five central tendency. They are:

(i) Arithmetic mean; (ii) Median; (iii) Mode; (iv) Geometric mean; (v) Harmonic mean; (vi) Quadratic mean

The Arithmetic Mean
The arithmetic mean, also popularly referred to as the ‘mean’, is the average of a series of figures or values. It is obtained by dividing the sum of these figures by the total number of the figures or values.

It is also the average of a collection of observation. The arithmetic mean is the most popularly used measure of central tendency.

Formula for calculating arithmetic mean
Arithmetic mean,   \overline{x}=\frac{\sum x}{n}

Where
\overline{x} = arithmetic mean
\sum x = the total of the values of series of figures in a given data
n = number of figure or elements.

Note; we use the formula \overline{x}=\frac{\sum x}{n}  to calculate arithmetic mean when number of figures (n) are small and ungrouped

Example
Calculate the arithmetic mean: 20, 10, 18, 5, 15, 35
Solution
Step 1: Add up the numbers/scores; \sum x  = 20 + 10 + 18 + 5 + 15 + 35 = 103
Step 2: Number of figures/scores = n = 6
Step 3: arithmetic mean =  \overline{x}=\frac{\sum x}{n}  = \frac{103}{6} = 17.2

There are situations where we encounter large repetitious data, then frequencies are used. Frequency is the number of times a particular event or information occurs. Frequency distribution is usually used when data are large and most of the numbers appear more than once, i.e. arithmetic formula for large data is \overline{x} = \frac{\sum fx}{n}  where f = number of times a particular number occur (frequency).

Example:
Calculate the mean of the following numbers; 20, 8, 10, 10, 20, 20, 30, 20, 8, 8, 5, 5, 5,
Solution:
Step 1: Arrange the numbers in ascending order to identify the numbers that occur in the set, i.e. 5, 5, 5, 8, 8, 8, 10, 10, 20, 20, 20, 20, 30
Step 2: Arrange the numbers in a frequency table as shown below:

Number (x) Frequency (f)
5 3
8 3
10 2
20 4
30 1

n = 13

i.e. \sum fx = (5 x 3) + (8 x 3) + (10 x 2) + (20 x 4) + (30 x 1) = 15 + 24 + 20 + 80 + 30 = 169.

n = 13

i.e. Arithmetic mean  \overline{x} = \frac{\sum fx}{n} = \frac{169}{13} = 13.

Advantages of Mean as a Statistical Tool
(i) It is highly representative of a given set. This is because all the values are considered.
(ii) it is easily understandable and easy to calculate.
(iii) Given that it is a calculated average, the mean is definite.
(iv) The mean lends itself more readily to further structural analysis when compared to other measures of central tendency, it exhibits higher degree of accuracy.

Disadvantages of Mean as a Statistical Tool
(i) When extreme scores are involved (either low or high), the mean is less reliable compared to median.
(ii) It is not readily depicted on frequency graphs.

Exercise
1. Calculate the mean of the following data: 56, 36, 62, 79, 83, 36, 62, 42, 62 and 42. (JAMB 2008)
2.

Days Bags of wheat produced daily
1 4
2 6
3 7
4 9
5 13

What is the mean of the data given above? (SSCE 1992)

 

Median
The median is that value which divides a set of numbers into two equal parts after arranging in ascending or descending order of magnitude.

Thus, the median can be determined by arranging a given set of numbers in either ascending or descending order of magnitude and picking the middle number if the sum of the set of numbers being considered is odd. If however, the set of numbers are even, the arithmetic mean of the two middle numbers will be determined to give the median of such a set.

How to calculate the median
(a)
When the numbers involved are odd numbers, the median will be the middle number.

Example: Find the median of the following sets of values: 20, 10, 18, 5, 15, 35, 40

Solution:
Step 1: Arrange the numbers in ascending or descending order of magnitude, i.e. we have 5, 10, 15, 18, 20, 35, 40. There are seven (7) numbers involved (odd number). Since there are 7 numbers involved, the middle number is the 4th number and the 4th number is 18.

Alternatively, you can use the median formula; i.e. median = \frac{n+1}{2} = \frac{7+1}{2}  =  \frac{8}{2}  = 4
i.e. the median is the 4th number which is 18.

(b) When even numbers are representing the number of events in data, the two middle values are taken; add them and divide them by two. The median will be the arithmetic mean of the two middle numbers.

Example: Find the median of the following sets of values: 20, 10, 18, 5, 15, 35

Solution:
Step 1: First arrange the numbers in ascending or descending order of magnitude, i.e. in ascending order, we have 5, 10, 15, 18, 20, 35
Step 2: Count the number of values involved. There are six (6) numbers involved, the middle number will be 3rd and 4th numbers which are 15 and 18
Step 3: To get the median, then add 15 + 18 and divide the answer by 2; i.e. \frac{15+18}{2} = 16.5

(c) When a group data is involved, cumulative frequency is used; i.e. for grouped data;

(i) median = \left ( \frac{n+1}{2} \right ) th when number of item (n) is odd

(ii) median = \frac{\left ( \frac{n}{2} \right )th + \left (\frac{n+1}{2} \right )th}{2}  when number of item (n) is even

Where n is the summation of all frequency and this is the terminal cumulative frequency.

Example: The median age of pupils in ASSURE school is represented in the table below:

Age (yrs) 4 5 6 7 8 9
Frequency 3 5 12 8 7 4

Solution:
Step 1: Draw a cumulative frequency table

Age (yrs) Frequency Cumulative Frequency
4 3 3
5 5 8
6 12 20
7 8 28
8 7 35
9 4 39

From the table above, there are 39 members as indicated by the terminal (last) cumulative frequency. Since the members are odd (39) the median age will be \left ( \frac{n+1}{2} \right )th member

i.e. median age = \left ( \frac{39+1}{2} \right )th = \left ( \frac{40}{2} \right )th = 20th member

i.e. the 20th falls within the cumulative under the age 6 years in the table above, therefore the median age is 6 years.

Example: The marks of pupils in ASSURE school is represented in a table as shown below.

Marks % 10 12 15 20 30 40
Frequency 4 6 7 5 8 10

Solution:
Step 1:
Construct a cumulative frequency table

Marks (%) Frequency Cumulative Frequency
10 4 4
12 6 10
15 7 17
20 5 22
30 8 40
40 10 50

From the table above, there are 50 members as indicated by the terminal (last) cumulative frequency. Since 50 is even, the median mark will be:  \frac{\left ( \frac{n}{2} \right )th + \left (\frac{n+1}{2} \right )th}{2}

Median = \frac{\left ( \frac{n}{2} \right )th + \left (\frac{n+1}{2} \right )th}{2}  = \frac{\left ( \frac{50}{2} \right )th + \left ( \frac{50+1}{2} \right )th}{2} = \frac{25th + 26th}{2}

The 25th member is 20 marks
The 26th member is 20 marks

i.e. median score = \left ( \frac{20+20}{2} \right ) = \left ( \frac{40}{2} \right ) = 20 marks

i.e. median score = 20 marks.

Advantages of Median as a Statistical Tool
(i) Easily understandable and easy to calculate.
(ii) Unlike the mean, it is not affected by extreme scores. That is, it is not sensitive to wide variations.
(iii) It can be depicted on frequency graphs.

Disadvantages of Median as a Statistical Tool
(i) When large numbers of scores are involved, arranging in ascending or descending order is a difficult and cumbersome task.
(ii) It is sensitive to small sample size.
(iii) It does not lend itself to further statistical analysis.
(iv) It is not representative of scores in a given set

Mode
This is the most recurring score in a given set of scores, items or events. In other words, it is the score or item with the highest frequency in a given distribution sample. The mode of a given set of scores is determined by inspecting carefully which of the items or scores occurs most number of times than other scores assuming the distribution is unimodal (having only one mode).

If two items or scores have the same number of highest occurrence, the distribution is considered to be bimodal. Further, if more than two items or scores have the most occurring frequency, such a distribution would be said to be multimodal.

For example, the mode of the following set of scores; 2, 3, 4, 4, 5, 5, and 5 will be 5 which occurred three times and more times than any other scores. If the distribution of a set of scores ¡s given as 2, 3, 3, 4, 4, 4 5, 5, 5, 6, 7, and 8, in this case 4 and 5 both occurred most frequently, three times each, this is a case of bimodal mode distribution.

Advantages of Mode as a Statistical Tool
(i) It is considerably simple to comprehend.
(ii) Also, it is easy to calculate.
(iii) It can be depicted graphically.
(iv) It is the most ideal, if there is the need for a quick measure of central tendency to be given in a set of scores.

Disadvantages of Mode as a Statistical Tool
(i) All items or scores in a given set of distribution are not duly considered. Thus, it may not be representative of a set of scores.
(ii) The possibility of bimodal and multimodal modes reduces the statistical usefulness of the mode.
(iii) The degree of accuracy determination in group data is poor.
(iv) Often, in view of (iii) above, its precise location in a given distribution is not definite.

Geometric Mean
Geometric mean formula, as the name suggests, is used to calculate the geometric mean of a set of numbers. Geometric mean (or GM) is a type of mean that indicates the central tendency of a set of numbers by using the product of their values.

It is defined as the nth root of the product of n numbers. It should be noted that you cannot calculate the geometric mean from the arithmetic mean. In statistics, the geometric mean is well defined only for a positive set of real numbers.

The geometric mean is denoted by letter GM. it is derived from a set of n observation by taking the nth root of the product of the numbers, i.e.

G = \sqrt[n]{x_{1}\times x_{2}\times x_{3}\times ......\times x_{n}}   where x = individual value

Example: Find the geometric mean of 4 and 3?
Solution: Using the formula for G.M., the geometric mean of 4 and 3 will be: \sqrt{4\times 3} = \sqrt{4} x \sqrt{3} = 2\sqrt{3} = 3.46
i.e. GM = 3.46

Example: Calculate the geometric mean of the following set of data: 2, 5, 7
Solution:
n = 3
product of the given values = 2 x 5 x 7 = 70
i.e. GM =\sqrt[3]{2\times 5\times 7} = \sqrt[3]{70}  = 4.063

Advantages of Geometric Mean
(i) It is rigidly defined.
(ii) It is based upon all the observations.
(iii) It is suitable for further mathematical treatment.
(iv) It is not affected much by fluctuations of samplings.
(v) It gives comparatively more weight to small items.

Disadvantages of Geometric Mean
(i) Because of its abstract mathematical character, geometric mean is not easy to understand and to calculate for non-mathematics person.
(ii) If any one of the observations is negative, geometric mean becomes imaginary regardless of the magnitude of the other items.

Harmonic mean
Harmonic mean is used to calculate the average of a set of numbers. The number of elements will be averaged and divided by the sum of the reciprocals of the elements.

It is calculated by dividing the number of observations by the sum of reciprocal of the observation.

The formula to find the harmonic mean is given by (for ungrouped data):

Hamomic Mean (HM) = \frac{n}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\frac{1}{x_{4}}+.....+ \frac{1}{x_{n}}}

Where n = total number of items

x1, x2, x3, x4,…….xn = individual terms or individual values

Example: Find the harmonic mean of the following data 6, 8, 9, 4, 5
Solution: Given data = 6, 8, 9, 4, 5; i.e. n = 5;
i.e. harmonic mean = \frac{5}{\frac{1}{6}+\frac{1}{8}+\frac{1}{9}+\frac{1}{4}+\frac{1}{5}} = \frac{5}{0.852}  = 5.868

i.e. HM = 5.868

Advantages of Harmonic mean
(i) Harmonic mean is rigidly defined, based upon all the observations and is suitable for further mathematical treatment. Like geometric mean it is not affected much by fluctuations of sampling.
(ii) It gives greater importance to small items and is useful only when small items have to be given a greater weightage.

Disadvantages of Harmonic mean
(i) Harmonic mean is not easily understood and is difficult to compute.

Quadratic Mean
Quadratic mean, also known as the root mean square (RMS), refers to the square root of the arithmetic mean of their squares. The quadratic mean is represented by Root Mean Square.

Quadratic Mean = \sqrt{\frac{\sum x^{2}}{n}}

Example: Calculate the quadratic mean of the following set of numbers: 3,4,7, 10
Solution:
Root Mean Square = \sqrt{\frac{\sum x^{2}}{n}} = \sqrt{\frac{3^{2}+4^{2}+7^{2}+10^{2}}{4}} = \sqrt{\frac{9+16+49+100}{4}} = \sqrt{\frac{174}{4}} =\sqrt{43.5}  = 6.5726

Advantages of quadratic mean
(i) All values in a given data are taken into consideration.
(ii) It is easy to determine.

Disadvantages of quadratic mean
(i) Calculation becomes very difficult when given values are large.
(ii) Its principles are difficult to understand.

 

Unit 4: Linear and Simultaneous Equations
Linear equation is an equation that contains more than one unknown quantities represented by ax + by = c where x and y are unknown and a,b,c are numbers. Simultaneous equations are two or more equations which contain common unknown quantities which can be solved at the same time.

Simultaneous equations can be
(i) All linear equation
(ii) One linear, one quadratic equation

Simultaneous equations can be solved using the following methods:
(i) Elimination method
(ii) Substitution method
(iii) Graphical method

Example: Solve x + y = 13 and x – y = 5 by elimination method.
Solution: to avoid confusion, always label the equation as eqn (i) and (ii)
x + y = 13…………..eqn (i)
x – y = 5……………..eqn (ii)

by adding equations (i) and (ii) we can eliminate one of the unknown terms, which will reduce to simple equation, after that, we substitute the value obtained into any of the two equations to get the value of the second unknown.
i.e. to eliminate y we add eqn (i) and eqn (ii)
x + y = 13…………..eqn (i)
x – y = 5……………..eqn (ii)
2x     = 18
Divide both sides by 2;
\frac{2x}{2} = \frac{18}{2}

i.e. x = 9
Substitute the value of x = 9 into eqn (i)
9 + y = 13
Subtract 9 from both sides
9 – 9 + y = 13 – 9
i.e. y = 4

Example: Solve x + y = 8 and x – y = 12 by substitution method.
Solution: the method of substitution involves solving one of the two linear equations for one of the unknown after which this solution is substituted for that term in the second equation.

x + y = 8……………..eqn(i)
x – y = 12…………….eqn(ii)
from equation (ii), make x the subject
x – y = 12
i.e. x – y + y = 12 + y
i.e. x = 12 + y…………….eqn(iii)
Substituting eqn(iii) into eqn(i)
x + y = 8
i.e. 12 + y + y = 8
i.e. 12 + 2y = 8
i.e. 12 – 8 = -2y
i.e. 4 = -2y
Divide both sides by 2
\frac{4}{2} = \frac{-2y}{2}

i.e. 2 = -y
i.e. y = -2
Substitute the value of y into eqn(iii)
x = 12 + (-2) = 12 – 2 = 10
i.e. x = 10

Word Problems Leading to Simultaneous Equations
To solve a word problem leading to an equation follow these steps:
(i) Decide the unknown quantity and represent it by a letter such as x
(ii) State clearly the units used when necessary and convert to the same unit.
(iii) From an equation to represent the facts provided by the problem
(iv) Solve the equation using a suitable method and use the solution to answer the question in words

Example: The sum of two numbers is 15 and their difference is 1. Find the relations between these numbers. Find the two numbers.
Solution:
Let a represent the 1st number
Let b represent the 2nd number
i.e.
a + b = 15…………..eqn(i)
a – b = 1……………eqn(ii)
Using elimination method, add eqn (i) and (ii)
i.e. 2a = 16
divide both sides by 2
\frac{2a}{2} = \frac{16}{2}
i.e. a = 8

Exercise
The sum of the ages of a boy and a girl is 43 years. If the difference of their ages is 3 years, calculate the ages of the boy and girl.

 

Unit 5: Measures of Dispersion or Variability
This is the degree to which the items or scores of a given statistical distribution are scattered about their mean. In other words, dispersion or variability measures the spread of items from one another in a statistical distribution. That is how far apart the scores in a set of scores or items are. There are several measures of dispersion with the range, variance and standard deviation being the most important and commonly used. .

The Range
The range of a given distribution is the difference between the highest score and the lowest score. Thus, it is very simple to calculate. It should be noted that while two distributions may have equal range, the degree of dispersion or variability in each distribution will be markedly different. This is a major weakness of range as a statistical tool as It is an unreliable measure of dispersion.

Example: Find the range of the following data: 10, 5, 9, 11, 15
Solution:

The maximum (highest) value = 15
The minimum (lowest) value = 5
The range = 15 – 5 = 10

Advantages of Range as a Statistical Tool
(i) The simplicity of calculation is a major advantage of range
(ii) As it stands, this statistics provides a quick indication of variability
(iii) Sample range as an estimator of the population range is easily understood.

Disadvantages of Range as a Statistical Tool
(i) Difficulty of interpretation is a major limitation.
(ii) Like arithmetic mean, sample range is highly sensitive to extreme large or small values.
(iii) It is not well defined for open-ended set of scores or population size.

 

Variance
This is expressed as the arithmetic mean of the square of the deviations of all items or scores in a given distribution of items or scores from their arithmetic mean that is, the expected value of the squared deviations of items from the mean. Thus, mathematically, the variance denoted

Variance is given by = \frac{\sum \left ( x_{i} -\overline{x}\right )^{2}}{n}

Where xi stand for each of the observation;  \overline{x} = arithmetic mean and n = number of items in the distribution

Example:
Calculate the variance of the following set of numbers: 3, 5, 8, 4
Solution:
Step 1
: Calculate the arithmetic mean,  \overline{x} = \frac{3+5+8+4}{4} = \frac{20}{4} = 5
Step 2: Calculate the deviation (xi  –  \overline{x})
i.e. |3 – 5|, |5 – 5|, |8 – 5|, |4 – 5|
i.e. -2, 0, -3, -1
Step 3: Calculate the squares of these deviations (xi  –  \overline{x})2
i.e. (-2)2, (0)2, (-3)2, (-1)2
i.e. 4, 0, 9, 1
Step 4: Add up all these squares; i.e. 4 + 0 + 9 + 1 = 14
Step 5: Calculate the arithmetic mean of the sum of the squares
i.e. Variance = \frac{\sum \left ( x_{i} -\overline{x}\right )^{2}}{n}= \frac{14}{4} = 3.5

Advantages of Variance as a Statistical Tool
(i) It is central to statistical computation as it lends itself to further use of statistical analysis.
(ii) It is a better measure of variability than the range.
(iii) It is widely adaptable for use in applied statistics .

Disadvantages of Variance as a Statistical Tool
(i) It is deficient in some certain positive mathematical properties.
(ii) The squaring of variations of a given data may impact on the true picture of the dispersion or variability.

 

Standard Deviation
This is the positive square-root of the sample variance. It is alternatively referred to as Root Mean Squared Deviation, for it is the mostly used measure of variability in applied statistics.

Example:
Calculate the standard deviation of the following set of numbers: 3, 5, 8, 4
Solution:
Step 1: Calculate the variance following the Step 1 to 5 described in the previous exercise.
Step 2: Calculate the standard deviation; i.e. standard deviation = \sqrt{variance}
i.e.  \sqrt{3.5} = 12.25

Exercises
1. Calculate the variance and standard deviation of the following set of numbers: 2, 5, 6, 7, 7, 9

Advantages of Standard Deviation as a Statistical Tool
(i) It is the most reliable estimate of viability measure, and thus, lends itself for use in other numerous statistical computations.
(ii) It is immensely useful in analysis of distribution of scores and making statistical deductions.
(iii) It has the advantage of being denominated in the same units of the original items unlike the sample variance that is denominated in squares of the original items.
(iv) It exhibits a good deal of positive mathematical properties; this is a reason for its being used in other statistical calculations.

Disadvantages of Standard Deviation as a Statistical Tool
(i) It lays more emphasis on the use of extreme values
(ii) Its calculation is very difficult and tedious.

End of SS1 Economics 2nd Term Lessons

 

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