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Compiled by: Susan Mburu
Descriptive statistics are essential in Cooperative Management for summarizing and interpreting data collected from members and operations. Understanding measures of central tendency enables cooperative managers to make informed decisions by identifying typical values within data sets such as member contributions, loan amounts, or sales volumes. This chapter focuses on the primary measures of central tendency: mean, mode, and median, providing practical computation techniques and worked examples relevant to cooperative settings in Kenya.
Measures of central tendency summarize a data set by identifying a central point around which data values cluster. In cooperative management, these measures help analyze financial transactions, member participation statistics, and product sales figures to determine average performance or common characteristics. The three main measures are mean, mode, and median, each with unique calculation methods and applications.
The mean, or arithmetic average, is the sum of all data values divided by the number of values. It is widely used in cooperatives to calculate average loan amounts disbursed, average savings per member, or average sales per period.
The formula for the mean is:
$$ \bar{x} = \frac{\Sigma x_i}{n} $$
where \( \bar{x} \) is the mean, \( \Sigma x_i \) is the sum of all data values, and \( n \) is the number of data points.
Example 1: A cooperative disbursed loans to 5 members of amounts Ksh 10,000, 15,000, 12,000, 18,000, and 20,000. Find the average loan amount.
Given: \(x_1 = 10000\), \(x_2 = 15000\), \(x_3 = 12000\), \(x_4 = 18000\), \(x_5 = 20000\), \(n=5\)
$$ \bar{x} = \frac{10000 + 15000 + 12000 + 18000 + 20000}{5} $$
$$ \bar{x} = \frac{75000}{5} $$
$$ \bar{x} = 15000 \text{ Ksh} $$
Answer: 15,000 Ksh
Example 2: A cooperative's monthly sales (in Ksh) for 6 months are: 250,000; 270,000; 260,000; 280,000; 300,000; 290,000. Calculate the mean monthly sales.
Given: \(x_1=250000\), \(x_2=270000\), \(x_3=260000\), \(x_4=280000\), \(x_5=300000\), \(x_6=290000\), \(n=6\)
$$ \bar{x} = \frac{250000 + 270000 + 260000 + 280000 + 300000 + 290000}{6} $$
$$ \bar{x} = \frac{1650000}{6} $$
$$ \bar{x} = 275000 \text{ Ksh} $$
Answer: 275,000 Ksh
Example 3: A cooperative recorded member savings (in Ksh) over 7 weeks as follows: 1200, 1500, 1100, 1400, 1300, 1600, 1250. Determine the average weekly savings.
Given: \(x_1=1200\), \(x_2=1500\), \(x_3=1100\), \(x_4=1400\), \(x_5=1300\), \(x_6=1600\), \(x_7=1250\), \(n=7\)
$$ \bar{x} = \frac{1200 + 1500 + 1100 + 1400 + 1300 + 1600 + 1250}{7} $$
$$ \bar{x} = \frac{9350}{7} $$
$$ \bar{x} = 1335.71 \text{ Ksh} $$
Answer: 1,335.71 Ksh
Example 4: A cooperative tracks the number of products sold daily for 8 days: 50, 65, 55, 70, 60, 75, 80, 68. Calculate the average daily sales.
Given: \(n=8\), \(x_i\) as above
$$ \bar{x} = \frac{50+65+55+70+60+75+80+68}{8} $$
$$ \bar{x} = \frac{523}{8} $$
$$ \bar{x} = 65.38 \text{ products} $$
Answer: 65.38 products
Example 5: The amount of dividends declared to 4 member groups are Ksh 5,000, 7,500, 6,000, and 8,000. Find the mean dividend per group.
Given: \(x_1=5000\), \(x_2=7500\), \(x_3=6000\), \(x_4=8000\), \(n=4\)
$$ \bar{x} = \frac{5000 + 7500 + 6000 + 8000}{4} $$
$$ \bar{x} = \frac{26500}{4} $$
$$ \bar{x} = 6625 \text{ Ksh} $$
Answer: 6,625 Ksh
The mode is the value that appears most frequently in a data set. In cooperative management, mode can identify the most common loan size requested or the most frequent number of products sold.
The mode is found by identifying the data value with the highest frequency.
Example 1: Loan amounts (in Ksh) given to members are: 10,000; 15,000; 10,000; 20,000; 10,000; 15,000. Find the mode.
Given data: \(10000, 15000, 10000, 20000, 10000, 15000\)
Frequency count:
Mode = 10,000 Ksh
Answer: 10,000 Ksh
Example 2: Daily sales quantities (units) for 7 days: 20, 25, 20, 30, 25, 20, 35. Find the mode.
Given data: \(20, 25, 20, 30, 25, 20, 35\)
Frequency count:
Mode = 20 units
Answer: 20 units
Example 3: Member savings (Ksh) in a week: 500, 600, 700, 600, 800, 600, 700. Find the mode.
Given data: \(500, 600, 700, 600, 800, 600, 700\)
Frequency count:
Mode = 600 Ksh
Answer: 600 Ksh
Example 4: Number of products sold per day over 10 days: 12, 15, 12, 18, 12, 20, 15, 18, 12, 15. Find the mode.
Given data: \(12, 15, 12, 18, 12, 20, 15, 18, 12, 15\)
Frequency count:
Mode = 12 units
Answer: 12 units
Example 5: Dividends declared (Ksh) to 6 groups: 4000, 5000, 4000, 6000, 5000, 4000. Find the mode.
Given data: \(4000, 5000, 4000, 6000, 5000, 4000\)
Frequency count:
Mode = 4000 Ksh
Answer: 4000 Ksh
The median is the middle value of an ordered data set, dividing it into two equal halves. It is useful in cooperative management to understand typical values when data contains outliers, such as unusually large loan amounts.
To find the median:
Example 1: Find the median of loan amounts (Ksh): 10,000; 15,000; 12,000; 18,000; 20,000.
Given data: \(10000, 15000, 12000, 18000, 20000\)
Step 1: Arrange in ascending order:
\(10000, 12000, 15000, 18000, 20000\)
Step 2: Number of data points \(n=5\) (odd), median is the 3rd value:
Median = 15,000 Ksh
Answer: 15,000 Ksh
Example 2: Find the median of monthly sales (Ksh) for 6 months: 250,000; 270,000; 260,000; 280,000; 300,000; 290,000.
Given data: \(250000, 270000, 260000, 280000, 300000, 290000\)
Step 1: Arrange in ascending order:
\(250000, 260000, 270000, 280000, 290000, 300000\)
Step 2: Number of data points \(n=6\) (even), median is average of 3rd and 4th values:
$$ \text{Median} = \frac{270000 + 280000}{2} $$
$$ \text{Median} = \frac{550000}{2} $$
$$ \text{Median} = 275000 \text{ Ksh} $$
Answer: 275,000 Ksh
Example 3: Find the median of weekly member savings (Ksh): 1200, 1500, 1100, 1400, 1300, 1600, 1250.
Given data: \(1200, 1500, 1100, 1400, 1300, 1600, 1250\)
Step 1: Arrange in ascending order:
\(1100, 1200, 1250, 1300, 1400, 1500, 1600\)
Step 2: Number of data points \(n=7\) (odd), median is 4th value:
Median = 1300 Ksh
Answer: 1300 Ksh
Example 4: Find the median of daily sales units over 8 days: 50, 65, 55, 70, 60, 75, 80, 68.
Given data: \(50, 65, 55, 70, 60, 75, 80, 68\)
Step 1: Arrange in ascending order:
\(50, 55, 60, 65, 68, 70, 75, 80\)
Step 2: Number of data points \(n=8\) (even), median is average of 4th and 5th values:
$$ \text{Median} = \frac{65 + 68}{2} $$
$$ \text{Median} = \frac{133}{2} $$
$$ \text{Median} = 66.5 \text{ units} $$
Answer: 66.5 units
Example 5: Find the median of dividends declared (Ksh) to 4 groups: 5,000; 7,500; 6,000; 8,000.
Given data: \(5000, 7500, 6000, 8000\)
Step 1: Arrange in ascending order:
\(5000, 6000, 7500, 8000\)
Step 2: Number of data points \(n=4\) (even), median is average of 2nd and 3rd values:
$$ \text{Median} = \frac{6000 + 7500}{2} $$
$$ \text{Median} = \frac{13500}{2} $$
$$ \text{Median} = 6750 \text{ Ksh} $$
Answer: 6,750 Ksh
A cooperative disbursed loans to 7 members: 12,000; 15,000; 10,000; 18,000; 20,000; 14,000; 16,000. Calculate the mean loan amount. (5 marks)
The following are daily sales (units) for 10 days: 30, 35, 30, 40, 35, 30, 45, 35, 30, 40. Find the mode of daily sales. (5 marks)
Member savings (Ksh) over 9 weeks are: 1,200; 1,100; 1,300; 1,400; 1,250; 1,500; 1,350; 1,200; 1,300. Calculate the median weekly savings. (5 marks)
Monthly dividend declared (Ksh) to 5 groups: 4,000; 4,500; 5,000; 4,000; 4,500. Find the mode. (5 marks)
A cooperative recorded the following sales volumes (units) over 6 days: 55, 60, 65, 70, 75, 80. Calculate the median sales volume. (5 marks)
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Create a free accountThis chapter covered key concepts in descriptive statistics essential for summarizing and interpreting data. It began with measures of central tendency, explaining how the mean provides the arithmetic average, the mode identifies the most frequent value, and the median represents the middle value in an ordered data set. The chapter then explored measures of dispersion, detailing how variance quantifies the average squared deviation from the mean and how standard deviation expresses this variability in the same units as the data. Lastly, the chapter introduced measures of skewness and kurtosis, which describe the asymmetry and the peakedness or flatness of a data distribution, respectively. Together, these statistical tools enable a comprehensive understanding of data patterns, variability, and distribution shape, crucial for informed decision-making in business contexts.
A cooperative recorded the weekly sales (in Ksh thousands) of 5 of its members as follows: 12, 15, 14, 16, 13. Calculate the mean sales. (2 marks)
The number of loan applications received by a SACCO over 7 days were: 3, 5, 4, 3, 6, 3, 5. Find the mode of the loan applications. (2 marks)
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