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Compiled by: Samuel Maina
Descriptive statistics are essential in Cooperative Management for summarizing and interpreting data collected from members, financial records, and operational activities. Measures of central tendency provide key insights into typical values within datasets, enabling cooperative managers to make informed decisions. This chapter focuses on the three primary measures of central tendency, mean, mode, and median, and their application in cooperative settings such as member contribution analysis, loan repayment patterns, and sales data interpretation.
Measures of central tendency summarize data by identifying a central or typical value, providing a snapshot that simplifies complex data sets. In cooperative management, understanding these measures aids in evaluating member behavior, financial performance, and resource allocation. The three main measures are mean, mode, and median, each appropriate for different data types and distributions.
The mean is the arithmetic average of a data set, calculated by summing all values and dividing by the number of observations. It is widely used in cooperative financial analysis, such as determining average member savings or average loan amounts.
$$\text{Mean} = \frac{\sum x_i}{n}$$
where \(x_i\) are the individual data points and \(n\) is the number of data points.
Example 1: A cooperative has 5 members who saved the following amounts in Ksh: 2,000; 3,000; 2,500; 4,000; 3,500. Find the mean savings.
Given: \(x_1 = 2000\), \(x_2 = 3000\), \(x_3 = 2500\), \(x_4 = 4000\), \(x_5 = 3500\), \(n=5\)
$$\text{Mean} = \frac{2000 + 3000 + 2500 + 4000 + 3500}{5}$$
$$= \frac{15000}{5}$$
$$= 3000 \text{ Ksh}$$
Answer: 3000 Ksh
Example 2: A SACCO recorded the loan amounts disbursed to 8 members as follows (in Ksh): 10,000; 15,000; 12,000; 18,000; 20,000; 16,000; 14,000; 17,000. Calculate the mean loan amount.
Given: \(n=8\), data points as above.
$$\text{Mean} = \frac{10000 + 15000 + 12000 + 18000 + 20000 + 16000 + 14000 + 17000}{8}$$
$$= \frac{122000}{8}$$
$$= 15250 \text{ Ksh}$$
Answer: 15,250 Ksh
Example 3: A cooperative’s monthly sales in Ksh over 6 months are: 50,000; 65,000; 55,000; 70,000; 60,000; 75,000. Find the mean monthly sales.
Given: \(n=6\), data points as above.
$$\text{Mean} = \frac{50000 + 65000 + 55000 + 70000 + 60000 + 75000}{6}$$
$$= \frac{375000}{6}$$
$$= 62500 \text{ Ksh}$$
Answer: 62,500 Ksh
Example 4: A cooperative wants to find the average number of products sold per day over 10 days: 120, 135, 140, 110, 125, 130, 150, 145, 115, 140. Calculate the mean.
Given: \(n=10\), data points as above.
$$\text{Mean} = \frac{120 + 135 + 140 + 110 + 125 + 130 + 150 + 145 + 115 + 140}{10}$$
$$= \frac{1310}{10}$$
$$= 131 \text{ products}$$
Answer: 131 products
Example 5: A cooperative’s annual dividend per member (in Ksh) for 7 members is: 800; 900; 850; 1000; 950; 1100; 1050. Find the mean dividend.
Given: \(n=7\), data points as above.
$$\text{Mean} = \frac{800 + 900 + 850 + 1000 + 950 + 1100 + 1050}{7}$$
$$= \frac{6650}{7}$$
$$= 950 \text{ Ksh}$$
Answer: 950 Ksh
The mode is the value that appears most frequently in a data set. It is especially useful in cooperative management for identifying the most common loan amount, membership category, or product sold. Mode can be used with nominal, ordinal, or numerical data.
Example 1: A cooperative records the number of shares held by 10 members: 100, 150, 100, 200, 150, 150, 200, 100, 150, 100. Find the mode.
Given: data points as above.
Since 100 and 150 both appear 4 times, the data is bimodal.
Answer: Modes are 100 and 150 shares
Example 2: The loan repayment periods (in months) for 9 members are: 12, 12, 18, 24, 12, 18, 24, 18, 18. Find the mode.
Given: data points as above.
Answer: Mode is 18 months
Example 3: A cooperative tracks the most common product sold daily over 7 days: maize, beans, maize, sugar, maize, sugar, beans. Determine the mode.
Given: data points as above.
Answer: Mode is maize
Example 4: Number of dependents per member in a cooperative: 3, 4, 3, 2, 3, 5, 4, 3, 2, 3. Find the mode.
Given: data points as above.
Answer: Mode is 3 dependents
Example 5: A cooperative’s monthly attendance at meetings over 8 months: 50, 60, 50, 55, 60, 60, 55, 50. Find the mode.
Given: data points as above.
Bimodal: 50 and 60
Answer: Modes are 50 and 60 attendees
The median is the middle value in an ordered data set and divides the data into two equal halves. It is particularly useful in cooperative management when data is skewed or contains outliers, such as income levels or loan amounts.
Example 1: Find the median of member savings (Ksh): 1,000; 3,000; 2,500; 4,000; 3,500.
Given: \(n=5\), data points unordered.
Step 1: Arrange in order: 1000, 2500, 3000, 3500, 4000
Step 2: Median is the middle value (3rd value):
Answer: Median is 3000 Ksh
Example 2: Loan amounts (Ksh) for 6 members: 10,000; 15,000; 12,000; 18,000; 20,000; 16,000.
Given: \(n=6\)
Step 1: Arrange: 10,000; 12,000; 15,000; 16,000; 18,000; 20,000
Step 2: Median is average of 3rd and 4th values:
$$\frac{15000 + 16000}{2} = \frac{31000}{2} = 15500$$
Answer: Median is 15,500 Ksh
Example 3: Monthly sales (Ksh) over 7 months: 50,000; 65,000; 55,000; 70,000; 60,000; 75,000; 80,000.
Given: \(n=7\)
Step 1: Arrange: 50,000; 55,000; 60,000; 65,000; 70,000; 75,000; 80,000
Step 2: Median is 4th value:
Answer: Median is 65,000 Ksh
Example 4: Number of products sold per day over 8 days: 120; 135; 140; 110; 125; 130; 150; 145.
Given: \(n=8\)
Step 1: Arrange: 110; 120; 125; 130; 135; 140; 145; 150
Step 2: Median is average of 4th and 5th values:
$$\frac{130 + 135}{2} = \frac{265}{2} = 132.5$$
Answer: Median is 132.5 products
Example 5: Annual dividend per member (Ksh) for 9 members: 800; 900; 850; 1000; 950; 1100; 1050; 900; 850.
Given: \(n=9\)
Step 1: Arrange: 800; 850; 850; 900; 900; 950; 1000; 1050; 1100
Step 2: Median is 5th value:
Answer: Median is 900 Ksh
A cooperative’s weekly savings (Ksh) by 7 members are: 1500, 2000, 1800, 2200, 2100, 1900, 1700. Calculate the mean savings. (5 marks)
The number of loans disbursed per month in a SACCO over 9 months is: 12, 15, 12, 14, 15, 15, 13, 12, 15. Find the mode. (5 marks)
Find the median of member contributions (Ksh) recorded as: 5000, 7000, 6500, 6000, 8000, 7500. (5 marks)
A cooperative’s daily sales (units) over 10 days are: 50, 55, 60, 65, 70, 75, 80, 85, 90, 95. Calculate the mean daily sales. (5 marks)
The repayment periods (months) for 11 loans are: 12, 18, 12, 24, 18, 12, 24, 18, 24, 12, 18. Find the mode. (5 marks)
Find the median of the following loan amounts (Ksh): 12000, 15000, 18000, 20000, 16000, 14000, 17000, 13000. (5 marks)
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Create a free accountThis chapter focused on descriptive statistics, starting with the measures of central tendency which include the mean, mode, and median. The mean provides the average value of a data set, the mode identifies the most frequently occurring value, and the median represents the middle value when data is ordered. Next, the chapter covered measures of dispersion, explaining variance as the average of squared deviations from the mean and standard deviation as the square root of variance, indicating data spread. Finally, the chapter introduced measures of skewness and kurtosis, which describe the asymmetry and peakedness of data distributions respectively. Together, these concepts provide a comprehensive understanding of data characteristics essential for business and statistical analysis.
A cooperative society records the monthly savings (in Ksh) of 5 members as 1200, 1500, 1800, 2000, and 1700. Calculate the mean monthly savings. (2 marks)
The weekly sales (in Ksh) of a cooperative retail shop for 7 weeks are: 4500, 4600, 4700, 4600, 4800, 4600, 4900. Determine the mode of the weekly sales. (2 marks)
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