BASIC TOOLS FOR ECONOMIC ANALYSIS (MEASURES OF CENTRAL TENDENCY USING GROUPED DATA) Lesson Note
This SS2 Economics lesson note covers BASIC TOOLS FOR ECONOMIC ANALYSIS (MEASURES OF CENTRAL TENDENCY USING GROUPED DATA) for First Term, Week 1.
Lesson Note
1. LEARNING OBJECTIVES
At the end of the lesson, students should be able to:
i). Explain the meaning of measures of central tendency.
ii). Identify mean, median and mode as measures of central tendency.
iii). Calculate the mean of grouped data.
iv). Calculate the median of grouped data.
v). Determine the mode of grouped data.
vi). State the uses of economic analysis
2. Meaning of Measures of Central Tendency
Measures of central tendency are statistical measures used to find a single value that represents or describes the central or typical value in a set of data. The three common measures are the mean, median and mode.
3. Arithmetic Mean of Grouped Data
The arithmetic mean is obtained by the addition of all items under observations and then divided by the total number of items been observed. For grouped data, the mean is calculated by multiplying each class midpoint by its frequency, adding the products and dividing by the total frequency.
Formula: Mean (x̄) = Σfx / Σf
Where: f = frequency, x = class midpoint, fx = frequency × midpoint.
Worked Example:
| Income (₦'000) | Frequency (f) | Midpoint (x) | Fx |
|---|---|---|---|
| 20–29 | 4 | 24.5 | 98 |
| 30–39 | 6 | 34.5 | 207 |
| 40–49 | 8 | 44.5 | 356 |
| 50–49 | 5 | 54.5 | 272.5 |
| 60–69 | 2 | 64.5 | 129 |
| Σf = 25 | Σfx = 1,062.5 |
Mean = 1,062.5
25
= 42.5
Therefore, the mean income is ₦42,500.
4. Median of Grouped Data
The median is the middle number in an observed list of data. It divides an ordered set of observations into two equal parts. For grouped data, the median is estimated using the median class—the class containing the (N/2)th observation.
Formula: Median = L + [(N/2 − CF) / f] × h
Where:
L = lower boundary of the median class
N = total frequency
CF = cumulative frequency before the median class
f = frequency of the median class
h = class width
Worked Example:
| Income (₦'000) | Frequency (f) | Cumulative Frequency |
|---|---|---|
| 20–29 | 4 | 4 |
| 30–39 | 6 | 10 |
| 40–49 | 8 | 18 |
| 50–59 | 5 | 23 |
| 60–69 | 2 | 25 |
N = 25, so N/2 = 12.5. The median class is 40–49 because its cumulative frequency is the first to exceed 12.5. Using L = 39.5, CF = 10, f = 8 and h = 10:
Median = 39.5 + [(12.5 − 10) / 8] × 10 = 42.625
Therefore, the median income is approximately ₦42,625.
5. Mode of Grouped Data
The mode is the most common number in a grouped data. In grouped data, the class with the highest frequency is called the modal class. The mode can be estimated using the formula below.
Mode = L + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h
Where:
L = lower boundary of the modal class
f₁ = frequency of the modal class
f₀ = frequency of the class immediately before the modal class
f₂ = frequency of the class immediately after the modal class
h = class width
Worked Example:
| Income (₦'000) | Frequency (f) | Cumulative Frequency |
|---|---|---|
| 20–29 | 4 | 4 |
| 30–39 | 6 | 10 |
| 40–49 | 8 | 18 |
| 50–59 | 5 | 23 |
| 60–69 | 2 | 25 |
From the table above, the modal class is 40–49 because it has the highest frequency of 8. Here, L = 39.5, f₁ = 8, f₀ = 6, f₂ = 5 and h = 10.
Mode = 39.5 + [(8 − 6) / (16 − 6 − 5)] × 10 = 43.5
Therefore, the estimated mode is ₦43,500.
6. Uses in Economic Analysis
Mean helps economists summarize large quantities of numerical data with one representative value.
Median is useful when extreme values may distort the average, such as income or wealth data.
Mode helps identify the most common or frequently occurring value or category.
The measures assist in comparing economic conditions among individuals, groups, regions or periods.
They are useful in planning, policy formulation, research and interpretation of economic data
Evaluation
1. Measure of central tendency statistically deals with _______ of a distribution
A. left values B. right values C. central values D. disarrayed values
2. When you have sum of items divided by the number of items, it is known as
A. mean B. median C. mode D. probability
3. The mode of a distribution is the
A. central value B. average value C. most frequent value D. arithmetic value
4. The three common measures of central tendency are
A. mean, median and mode B. mean, median and range C. mean, median and probability D. mode, range and probability
5. Which of these divides an observed data into two equal parts?
A. Mean B. Median C. Mode D. probability
6. The chance that an event will occur is known as
A. Mean B. range C. probability D. Mode
7. The difference between the highest value and the lowest value is the
A. range B. probability C. median D. mode
USE THE INFORMATION TO ANSWER QUESTIONS 8 – 10
5, 4, 3, 5, 5, 4, 4, 1, 2, 0, 6, 2, 2, 3, 2
8. The mean of the distribution is
A. 4.8 B. 3.5 C. 3.2 D. 2.3
9. The median of the distribution
A. 0 B. 3 C. 4 D. 5
10. The mode of the distribution is
A. 2 B. 3 C. 4 D. 5
Theory Questions
Worked Example:
Income (₦'000) : Frequency (f)
40–49 : 6
50–59 : 8
60–69 : 10
70–79 : 7
80–89 : 4
Find i) mean ii) median iii) mode