This chapter focuses on descriptive statistics essential for procurement management professionals in Kenya. Understanding and applying measures of central tendency enables procurement officers to summarise and interpret data such as supplier prices, delivery times, and purchase quantities effectively. These measures help in making informed decisions by providing representative values from large datasets encountered in procurement processes.

5.1 Measures of central tendency

Measures of central tendency summarise data sets by identifying a central or typical value. In procurement management, these statistics help analyse price quotations, supplier performance, and demand forecasts to ensure optimal purchasing decisions. The three primary measures are the mean, mode, and median, each providing unique insights into the data.

5.1.1 Mean

The mean or average is the sum of all the data values divided by the number of values. It provides a balanced central value and is widely used in procurement to assess average costs, delivery times, or quantities ordered.

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 values.

Worked Examples

Example 1: Calculate the mean price of 5 quotations for office chairs (in Ksh): 12,000; 13,500; 11,800; 12,600; 13,200.

Given: Prices = 12,000, 13,500, 11,800, 12,600, 13,200; \(n = 5\)

$$\bar{x} = \frac{12,000 + 13,500 + 11,800 + 12,600 + 13,200}{5}$$

$$\bar{x} = \frac{63,100}{5}$$

$$\bar{x} = 12,620 \text{ Ksh}$$

Answer: 12,620 Ksh

Example 2: A procurement officer records delivery times (in days) from 6 suppliers: 5, 7, 4, 6, 8, 5. Find the mean delivery time.

Given: Delivery times = 5, 7, 4, 6, 8, 5; \(n = 6\)

$$\bar{x} = \frac{5 + 7 + 4 + 6 + 8 + 5}{6}$$

$$\bar{x} = \frac{35}{6}$$

$$\bar{x} = 5.83 \text{ days}$$

Answer: 5.83 days

Example 3: Calculate the mean quantity ordered of cement bags over 8 months: 120, 135, 140, 110, 115, 125, 130, 128 bags.

Given: Quantities = 120, 135, 140, 110, 115, 125, 130, 128; \(n = 8\)

$$\bar{x} = \frac{120 + 135 + 140 + 110 + 115 + 125 + 130 + 128}{8}$$

$$\bar{x} = \frac{1003}{8}$$

$$\bar{x} = 125.38 \text{ bags}$$

Answer: 125.38 bags

Example 4: A procurement team records the cost (in Ksh) of 7 different suppliers’ bids for a printing contract: 80,000; 85,500; 79,000; 82,500; 81,000; 84,000; 83,000. Calculate the mean bid price.

Given: Bids = 80,000; 85,500; 79,000; 82,500; 81,000; 84,000; 83,000; \(n=7\)

$$\bar{x} = \frac{80,000 + 85,500 + 79,000 + 82,500 + 81,000 + 84,000 + 83,000}{7}$$

$$\bar{x} = \frac{575,000}{7}$$

$$\bar{x} = 82,142.86 \text{ Ksh}$$

Answer: 82,142.86 Ksh

Example 5: The delivery weights (in tonnes) from 5 suppliers are: 2.5, 3.0, 2.8, 3.2, 2.7. Find the mean delivery weight.

Given: Weights = 2.5, 3.0, 2.8, 3.2, 2.7; \(n=5\)

$$\bar{x} = \frac{2.5 + 3.0 + 2.8 + 3.2 + 2.7}{5}$$

$$\bar{x} = \frac{14.2}{5}$$

$$\bar{x} = 2.84 \text{ tonnes}$$

Answer: 2.84 tonnes

5.1.2 Mode

The mode is the value that occurs most frequently in a data set. In procurement, the mode helps identify the most common supplier price, delivery time, or quantity ordered, useful for understanding typical conditions or preferences.

No formula applies directly, but the mode is identified by frequency counting.

Worked Examples

Example 1: Find the mode of delivery times (in days) from suppliers: 5, 7, 5, 6, 8, 5.

Given: Delivery times = 5, 7, 5, 6, 8, 5

• Frequency: 5 appears 3 times, 6 once, 7 once, 8 once

Answer: 5 days

Example 2: Find the mode of bid prices (in Ksh) from 7 suppliers: 90,000; 85,000; 90,000; 88,000; 85,000; 90,000; 87,000.

Given: Bids = 90,000; 85,000; 90,000; 88,000; 85,000; 90,000; 87,000

• Frequency: 90,000 occurs 3 times, 85,000 twice, others once

Answer: 90,000 Ksh

Example 3: Identify the mode of quantities ordered (in units): 10, 15, 10, 20, 15, 10, 25.

Given: Quantities = 10, 15, 10, 20, 15, 10, 25

• Frequency: 10 appears 3 times, 15 twice, others once

Answer: 10 units

Example 4: Determine the mode of supplier ratings (scale 1-5): 4, 5, 4, 3, 4, 2, 5.

Given: Ratings = 4, 5, 4, 3, 4, 2, 5

• Frequency: 4 appears 3 times, 5 twice, others once

Answer: 4

Example 5: Find the mode of delivery delays (in days): 2, 3, 2, 4, 3, 2, 5.

Given: Delays = 2, 3, 2, 4, 3, 2, 5

• Frequency: 2 appears 3 times, 3 twice, others once

Answer: 2 days

5.1.3 Median

The median is the middle value when data are arranged in ascending order. It is useful in procurement for understanding the midpoint of prices or delivery times, especially when data have outliers.

To find the median:

• Arrange data in order
• If \(n\) is odd, median is the middle value
• If \(n\) is even, median is the average of the two middle values

Worked Examples

Example 1: Find the median price of bids (Ksh): 12,000; 13,500; 11,800; 12,600; 13,200.

Given: Prices = 12,000; 13,500; 11,800; 12,600; 13,200

• Ordered: 11,800; 12,000; 12,600; 13,200; 13,500

• \(n=5\) (odd), median = 3rd value = 12,600

Answer: 12,600 Ksh

Example 2: Find the median delivery time (days) from: 5, 7, 4, 6, 8, 5.

Given: Times = 5, 7, 4, 6, 8, 5

• Ordered: 4; 5; 5; 6; 7; 8

• \(n=6\) (even), median = average of 3rd and 4th values

$$\text{Median} = \frac{5 + 6}{2} = 5.5$$

Answer: 5.5 days

Example 3: Calculate the median quantity ordered (bags): 120, 135, 140, 110, 115, 125, 130, 128.

Given: Quantities = 120, 135, 140, 110, 115, 125, 130, 128

• Ordered: 110; 115; 120; 125; 128; 130; 135; 140

• \(n=8\) (even), median = average of 4th and 5th values

$$\text{Median} = \frac{125 + 128}{2} = 126.5$$

Answer: 126.5 bags

Example 4: Find the median supplier bid prices (Ksh): 80,000; 85,500; 79,000; 82,500; 81,000; 84,000; 83,000.

Given: Bids = 80,000; 85,500; 79,000; 82,500; 81,000; 84,000; 83,000

• Ordered: 79,000; 80,000; 81,000; 82,500; 83,000; 84,000; 85,500

• \(n=7\) (odd), median = 4th value = 82,500

Answer: 82,500 Ksh

Example 5: Calculate the median delivery weights (tonnes): 2.5, 3.0, 2.8, 3.2, 2.7.

Given: Weights = 2.5, 3.0, 2.8, 3.2, 2.7

• Ordered: 2.5; 2.7; 2.8; 3.0; 3.2

• \(n=5\) (odd), median = 3rd value = 2.8

Answer: 2.8 tonnes

Chapter Summary

This chapter focused on descriptive statistics, beginning with measures of central tendency which include the mean, mode, and median as key indicators of data center points. The mean provides the arithmetic average, the mode identifies the most frequently occurring value, and the median indicates the middle value in an ordered dataset. The chapter then covered measures of dispersion, emphasizing variance and standard deviation as tools to quantify data spread around the mean. Variance measures the average squared deviation, while standard deviation is the square root of variance, offering a more interpretable scale of variability. Finally, the chapter introduced measures of skewness and kurtosis, which describe the shape of the data distribution. Skewness indicates the degree of asymmetry, whereas kurtosis reflects the peakedness or flatness relative to a normal distribution. Together, these descriptive statistics provide a comprehensive summary of data characteristics essential for analysis in various fields.