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Descriptive statistics are fundamental in the accountancy profession for summarizing and interpreting financial data. Accountants in Kenya regularly analyze various financial figures such as sales, expenses, and returns to make informed decisions. This chapter focuses on measures of central tendency, which provide key insights into the typical values within data sets, aiding in budgeting, forecasting, and financial reporting.
Measures of central tendency are numerical values that represent the center or typical value of a data set. In accounting, these measures help summarize large amounts of financial data into a single representative figure, facilitating easier comparison and analysis. This section covers four primary measures: mean (arithmetic, weighted, geometric), harmonic mean, mode, and median, each with distinct applications in financial contexts.
The mean is the average value of a data set, widely used in accounting to determine average costs, revenues, or returns. The arithmetic mean is the simple average, while the weighted arithmetic mean accounts for varying importance of data points. The geometric mean is useful for calculating average growth rates, such as interest or inflation rates over multiple periods.
The arithmetic mean is calculated by summing all data values and dividing by the total number of values. It is the most common average used in accounting to assess typical financial figures.
$$\text{Arithmetic Mean} = \frac{\Sigma x_i}{n}$$
where \(x_i\) are the data values and \(n\) is the number of values.
Example 1: Calculate the average monthly sales (in Ksh) for a retail shop with sales: 120,000; 150,000; 130,000; 140,000; 160,000.
Given: \(x_1 = 120,000\), \(x_2 = 150,000\), \(x_3 = 130,000\), \(x_4 = 140,000\), \(x_5 = 160,000\), \(n = 5\)
$$\text{Arithmetic Mean} = \frac{120,000 + 150,000 + 130,000 + 140,000 + 160,000}{5}$$
$$= \frac{700,000}{5}$$
$$= 140,000 \text{ Ksh}$$
Answer: 140,000 Ksh
Example 2: An accountant records monthly expenses for 6 months as follows (in Ksh): 45,000; 55,000; 50,000; 60,000; 65,000; 70,000. Find the average monthly expense.
Given: \(x_1=45,000\), \(x_2=55,000\), \(x_3=50,000\), \(x_4=60,000\), \(x_5=65,000\), \(x_6=70,000\), \(n=6\)
$$\text{Arithmetic Mean} = \frac{45,000 + 55,000 + 50,000 + 60,000 + 65,000 + 70,000}{6}$$
$$= \frac{345,000}{6}$$
$$= 57,500 \text{ Ksh}$$
Answer: 57,500 Ksh
Example 3: A cooperative’s dividends for four years are: 8%, 10%, 12%, and 15%. Find the average dividend rate.
Given: \(x_1=8\%\), \(x_2=10\%\), \(x_3=12\%\), \(x_4=15\%\), \(n=4\)
$$\text{Arithmetic Mean} = \frac{8 + 10 + 12 + 15}{4}$$
$$= \frac{45}{4}$$
$$= 11.25\%$$
Answer: 11.25%
The weighted mean accounts for the relative importance or frequency of each value, commonly used in accounting when different transactions impact totals unequally.
$$\text{Weighted Mean} = \frac{\Sigma w_i x_i}{\Sigma w_i}$$
where \(w_i\) are weights and \(x_i\) data values.
Example 1: Calculate the average cost price of goods where quantities and unit costs are:
Given: \(w_1=100\), \(x_1=50\); \(w_2=200\), \(x_2=55\); \(w_3=150\), \(x_3=60\)
$$\text{Weighted Mean} = \frac{(100 \times 50) + (200 \times 55) + (150 \times 60)}{100 + 200 + 150}$$
$$= \frac{5,000 + 11,000 + 9,000}{450}$$
$$= \frac{25,000}{450}$$
$$= 55.56 \text{ Ksh}$$
Answer: 55.56 Ksh
Example 2: A bank calculates average interest rate on loans with amounts and rates:
Given: \(w_1=1,000,000\), \(x_1=10\%\); \(w_2=2,000,000\), \(x_2=12\%\); \(w_3=1,500,000\), \(x_3=11\%\)
$$\text{Weighted Mean} = \frac{(1,000,000 \times 10) + (2,000,000 \times 12) + (1,500,000 \times 11)}{1,000,000 + 2,000,000 + 1,500,000}$$
$$= \frac{10,000,000 + 24,000,000 + 16,500,000}{4,500,000}$$
$$= \frac{50,500,000}{4,500,000}$$
$$= 11.22\%$$
Answer: 11.22%
Example 3: A SACCO has members contributing shares as follows:
Calculate the average share price.
Given: \(w_1=300\), \(x_1=20\); \(w_2=500\), \(x_2=18\); \(w_3=200\), \(x_3=22\)
$$\text{Weighted Mean} = \frac{(300 \times 20) + (500 \times 18) + (200 \times 22)}{300 + 500 + 200}$$
$$= \frac{6,000 + 9,000 + 4,400}{1,000}$$
$$= \frac{19,400}{1,000}$$
$$= 19.4 \text{ Ksh}$$
Answer: 19.4 Ksh
The geometric mean is the average rate of return over multiple periods, commonly used in accountancy for compound interest and growth rates.
$$\text{Geometric Mean} = (\prod_{i=1}^n x_i)^{\frac{1}{n}}$$
where \(x_i\) are positive data values.
Example 1: Calculate the average growth rate of revenue over 3 years with rates: 5%, 10%, 15%.
Given: \(x_1=1.05\), \(x_2=1.10\), \(x_3=1.15\), \(n=3\)
$$\text{Geometric Mean} = (1.05 \times 1.10 \times 1.15)^{\frac{1}{3}}$$
$$= (1.32825)^{\frac{1}{3}}$$
$$= 1.100$$
$$= 1.100 - 1 = 0.100 = 10\%$$
Answer: 10% average growth rate
Example 2: A cooperative’s profit growth rates over 4 years are 8%, 12%, 10%, and 15%. Find the geometric mean.
Given: \(x_1=1.08\), \(x_2=1.12\), \(x_3=1.10\), \(x_4=1.15\), \(n=4\)
$$\text{Geometric Mean} = (1.08 \times 1.12 \times 1.10 \times 1.15)^{\frac{1}{4}}$$
$$= (1.533)^{\frac{1}{4}}$$
$$= 1.113$$
$$= 1.113 - 1 = 0.113 = 11.3\%$$
Answer: 11.3% average growth rate
Example 3: Calculate the average inflation rate over 5 years with rates: 6%, 7%, 5%, 8%, 9%.
Given: \(x_1=1.06\), \(x_2=1.07\), \(x_3=1.05\), \(x_4=1.08\), \(x_5=1.09\), \(n=5\)
$$\text{Geometric Mean} = (1.06 \times 1.07 \times 1.05 \times 1.08 \times 1.09)^{\frac{1}{5}}$$
$$= (1.347)^{\frac{1}{5}}$$
$$= 1.061$$
$$= 1.061 - 1 = 0.061 = 6.1\%$$
Answer: 6.1% average inflation rate
The harmonic mean is a special average useful when dealing with rates or ratios, such as average cost per unit or efficiency ratios in accountancy. It is calculated as the reciprocal of the arithmetic mean of the reciprocals of the data values.
$$\text{Harmonic Mean} = \frac{n}{\Sigma \frac{1}{x_i}}$$
where \(x_i\) are the data values and \(n\) is the number of values.
Example 1: Calculate the average cost per item when costs per unit are Ksh 20, 25, and 30.
Given: \(x_1=20\), \(x_2=25\), \(x_3=30\), \(n=3\)
$$\text{Harmonic Mean} = \frac{3}{\frac{1}{20} + \frac{1}{25} + \frac{1}{30}}$$
$$= \frac{3}{0.05 + 0.04 + 0.0333}$$
$$= \frac{3}{0.1233}$$
$$= 24.33 \text{ Ksh}$$
Answer: 24.33 Ksh
Example 2: A company processes invoices at rates of 15, 20, and 30 invoices per hour on three machines. Find the average processing rate per machine.
Given: \(x_1=15\), \(x_2=20\), \(x_3=30\), \(n=3\)
$$\text{Harmonic Mean} = \frac{3}{\frac{1}{15} + \frac{1}{20} + \frac{1}{30}}$$
$$= \frac{3}{0.0667 + 0.05 + 0.0333}$$
$$= \frac{3}{0.15}$$
$$= 20 \text{ invoices/hour}$$
Answer: 20 invoices/hour
Example 3: Calculate the average rate of return when rates are 12%, 15%, and 18%.
Given: \(x_1=12\), \(x_2=15\), \(x_3=18\), \(n=3\)
$$\text{Harmonic Mean} = \frac{3}{\frac{1}{12} + \frac{1}{15} + \frac{1}{18}}$$
$$= \frac{3}{0.0833 + 0.0667 + 0.0556}$$
$$= \frac{3}{0.2056}$$
$$= 14.59\%$$
Answer: 14.59%
The mode is the value that appears most frequently in a data set. Accountants use mode to identify the most common transaction amounts or frequently occurring financial values.
Mode is the data value with the highest frequency. Unlike mean and median, it does not have a formula but is identified by counting occurrences.
Example 1: Find the mode of sales figures (in Ksh): 120,000; 130,000; 120,000; 150,000; 120,000; 140,000.
Data frequencies: 120,000 (3 times), 130,000 (1), 140,000 (1), 150,000 (1).
Answer: 120,000 Ksh
Example 2: Identify the mode of invoice amounts (in Ksh): 5000; 7000; 5000; 6000; 7000; 7000.
Frequencies: 5000 (2), 6000 (1), 7000 (3)
Answer: 7000 Ksh
Example 3: A bank records the following loan amounts (in Ksh): 1,000,000; 1,200,000; 1,000,000; 1,500,000; 1,200,000; 1,200,000.
Frequencies: 1,000,000 (2), 1,200,000 (3), 1,500,000 (1)
Answer: 1,200,000 Ksh
When data is grouped in intervals, mode is estimated by:
$$\text{Mode} = L + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times h$$
where
Example 4: Given class intervals and frequencies:
| Class Interval (Ksh) | Frequency |
|---|---|
| 10,000 - 20,000 | 5 |
| 20,000 - 30,000 | 12 |
| 30,000 - 40,000 | 8 |
| 40,000 - 50,000 | 6 |
Find the mode.
Given: \(L=20,000\), \(f_1=12\), \(f_0=5\), \(f_2=8\), \(h=10,000\)
$$\text{Mode} = 20,000 + \frac{(12 - 5)}{(2 \times 12 - 5 - 8)} \times 10,000$$
$$= 20,000 + \frac{7}{(24 - 13)} \times 10,000$$
$$= 20,000 + \frac{7}{11} \times 10,000$$
$$= 20,000 + 6,363.64$$
Answer: 26,364 Ksh (approx.)
The median is the middle value in an ordered data set, splitting the data into two equal halves. It is especially useful in accounting when data is skewed, such as income or expense distributions.
To find the median, arrange data in ascending order and locate the middle value.
Example 1: Find the median of monthly sales (Ksh): 120,000; 140,000; 130,000; 150,000; 160,000.
Ordered data: 120,000; 130,000; 140,000; 150,000; 160,000
\(n=5\) (odd)
Median position = \(\frac{5+1}{2} = 3\)
Answer: 140,000 Ksh
Example 2: Calculate the median of expenses (Ksh): 45,000; 55,000; 50,000; 60,000.
Ordered data: 45,000; 50,000; 55,000; 60,000
\(n=4\) (even)
Median = \(\frac{50,000 + 55,000}{2} = 52,500 \text{ Ksh}\)
Answer: 52,500 Ksh
Example 3: A company’s quarterly profits (Ksh) are: 200,000; 180,000; 225,000; 210,000; 190,000; 205,000.
Ordered data: 180,000; 190,000; 200,000; 205,000; 210,000; 225,000
\(n=6\) (even)
Median = \(\frac{200,000 + 205,000}{2} = 202,500 \text{ Ksh}\)
Answer: 202,500 Ksh
Median class is identified as the class where cumulative frequency reaches or exceeds \(\frac{n}{2}\).
Median formula:
$$\text{Median} = L + (\frac{\frac{n}{2} - F}{f}) \times h$$
where
Example 4: Given class intervals and frequencies:
| Class Interval (Ksh) | Frequency |
|---|---|
| 10,000 - 20,000 | 5 |
| 20,000 - 30,000 | 12 |
| 30,000 - 40,000 | 8 |
| 40,000 - 50,000 | 5 |
Total \(n = 5 + 12 + 8 + 5 = 30\)
Cumulative frequencies:
Find median class (where cumulative frequency ≥ 15):
Median class = 20,000 - 30,000
Given: \(L=20,000\), \(n=30\), \(F=5\), \(f=12\), \(h=10,000\)
$$\text{Median} = 20,000 + (\frac{15 - 5}{12}) \times 10,000$$
$$= 20,000 + \frac{10}{12} \times 10,000$$
$$= 20,000 + 8,333.33$$
Answer: 28,333 Ksh (approx.)
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Create a free accountThis chapter covered key descriptive statistics used in business mathematics and statistics. It began with measures of central tendency, including arithmetic mean, weighted arithmetic mean, geometric mean, harmonic mean, mode, and median, explaining how each summarizes data. The discussion then moved to measures of dispersion such as range, quartiles, deciles, percentiles, mean deviation, standard deviation, and coefficient of variation, which describe the spread or variability of data sets. Finally, the chapter introduced measures of skewness and kurtosis, focusing on their roles in assessing data distribution shapes without involving coefficient calculations. Together, these concepts provide a comprehensive foundation for summarizing and interpreting business data effectively.
A firm records the following monthly sales (in Ksh 000): 120, 150, 130, 170, 160. Calculate the arithmetic mean of the sales. (3 marks)
An accountant has data on five employees’ working hours and their corresponding hourly wages:
| Employee | Hours Worked | Hourly Wage (Ksh) |
|---|---|---|
| A | 40 | 500 |
| B | 35 | 450 |
| C | 45 | 550 |
| D | 50 | 600 |
| E | 30 | 400 |
Calculate the weighted arithmetic mean hourly wage. (4 marks)
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