5.1 Measures of central tendency

Measures of central tendency summarize a dataset by identifying a single value that represents the center or typical value of the data. In procurement management, these measures can help analyze supplier bids, delivery times, or quality scores to guide decision-making. The three common measures are mean, mode, and median, each with distinct calculation methods and applications.

5.1.1 Mean

The mean or average is the sum of all data values divided by the number of values. It is useful in procurement for calculating average cost, average delivery time, or average quantity supplied. The formula for the mean of \(n\) values \(x_1, x_2, ..., x_n\) is:

$$ \bar{x} = \frac{\Sigma x_i}{n} $$

Worked Examples

Example 1:
Calculate the mean unit price of five bids received for a supply contract. The prices (in Ksh) are 120,000; 135,000; 125,000; 130,000; and 140,000.
Given: \(x_1=120,000\), \(x_2=135,000\), \(x_3=125,000\), \(x_4=130,000\), \(x_5=140,000\), \(n=5\)$$ \bar{x} = \frac{120,000 + 135,000 + 125,000 + 130,000 + 140,000}{5} $$$$ \bar{x} = \frac{650,000}{5} $$$$ \bar{x} = 130,000 \text{ Ksh} $$ Answer: 130,000 Ksh

Example 2:
A supplier delivers goods in 6 batches with quantities (units): 200, 180, 220, 210, 190, 200. Find the mean quantity delivered.
Given: \(x_1=200\), \(x_2=180\), \(x_3=220\), \(x_4=210\), \(x_5=190\), \(x_6=200\), \(n=6\)$$ \bar{x} = \frac{200 + 180 + 220 + 210 + 190 + 200}{6} $$$$ \bar{x} = \frac{1200}{6} $$$$ \bar{x} = 200 \text{ units} $$ Answer: 200 units

Example 3:
The delivery times (in days) from five suppliers are 7, 9, 8, 10, and 6 days. Calculate the mean delivery time.
Given: \(x_1=7\), \(x_2=9\), \(x_3=8\), \(x_4=10\), \(x_5=6\), \(n=5\)$$ \bar{x} = \frac{7 + 9 + 8 + 10 + 6}{5} $$$$ \bar{x} = \frac{40}{5} $$$$ \bar{x} = 8 \text{ days} $$ Answer: 8 days

Example 4:
A procurement officer records monthly expenditure (Ksh in thousands) for six months: 150, 165, 155, 160, 170, 175. Find the mean expenditure.
Given: \(x_1=150\), \(x_2=165\), \(x_3=155\), \(x_4=160\), \(x_5=170\), \(x_6=175\), \(n=6\)$$ \bar{x} = \frac{150 + 165 + 155 + 160 + 170 + 175}{6} $$$$ \bar{x} = \frac{975}{6} $$$$ \bar{x} = 162.5 \text{ Ksh thousands} $$ Answer: 162.5 Ksh thousands

Example 5:
The quantity of items ordered in 7 procurement requests is: 50, 60, 55, 65, 70, 60, 75 units. Calculate the mean quantity ordered.
Given: \(x_1=50\), \(x_2=60\), \(x_3=55\), \(x_4=65\), \(x_5=70\), \(x_6=60\), \(x_7=75\), \(n=7\)$$ \bar{x} = \frac{50 + 60 + 55 + 65 + 70 + 60 + 75}{7} $$$$ \bar{x} = \frac{435}{7} $$$$ \bar{x} = 62.14 \text{ units} $$ Answer: 62.14 units

5.1.2 Mode

The mode is the value that appears most frequently in a data set. In procurement, mode can identify the most common bid price, frequently ordered quantity, or typical delivery period. It is useful when the most popular or frequent value is important.

Worked Examples

Example 1:
Find the mode of supplier bid prices (Ksh): 120,000; 125,000; 120,000; 130,000; 125,000; 120,000.
Given data set: 120,000; 125,000; 120,000; 130,000; 125,000; 120,000
The frequency of each price:

• 120,000 appears 3 times
• 125,000 appears 2 times
• 130,000 appears 1 time
  Answer: 120,000 Ksh (mode)

Example 2:
Delivery days recorded: 7, 9, 8, 7, 10, 7, 9. Find the mode.
Given data set: 7, 9, 8, 7, 10, 7, 9
Frequencies:

• 7 appears 3 times
• 9 appears 2 times
• 8 appears 1 time
• 10 appears 1 time
  Answer: 7 days (mode)

Example 3:
Quantities ordered (units): 100, 110, 120, 110, 130, 110, 140. Determine the mode.
Given data set: 100, 110, 120, 110, 130, 110, 140
Frequencies:

• 110 appears 3 times
• Others appear once each
  Answer: 110 units (mode)

Example 4:
Prices (Ksh) of items purchased: 2000, 2200, 2000, 2300, 2200, 2200, 2400. Find the mode.
Given data set: 2000, 2200, 2000, 2300, 2200, 2200, 2400
Frequencies:

• 2200 appears 3 times
• 2000 appears 2 times
• 2300 and 2400 appear once each
  Answer: 2200 Ksh (mode)

Example 5:
Number of days taken to deliver orders: 15, 20, 15, 18, 15, 20, 22. Determine the mode.
Given data set: 15, 20, 15, 18, 15, 20, 22
Frequencies:

• 15 appears 3 times
• 20 appears 2 times
• 18 and 22 appear once each
  Answer: 15 days (mode)

5.1.3 Median

The median is the middle value when data is arranged in ascending or descending order. It divides the data into two equal halves and is useful in procurement to understand the central tendency when data is skewed by outliers. If the number of observations is even, the median is the average of the two middle values.

Worked Examples

Example 1:
Find the median of bid prices (Ksh): 120,000; 130,000; 125,000; 135,000; 140,000.
Given data: 120,000; 130,000; 125,000; 135,000; 140,000
Step 1: Order data ascending: 120,000; 125,000; 130,000; 135,000; 140,000
Step 2: Number of data points \(n=5\) (odd)
Step 3: Median is the middle value, position \( \frac{n+1}{2} = \frac{6}{2} = 3 \)
Step 4: Median = 130,000 Ksh
Answer: 130,000 Ksh

Example 2:
Delivery days from 6 suppliers: 8, 7, 9, 6, 10, 7. Find the median.
Given data: 8, 7, 9, 6, 10, 7
Step 1: Order ascending: 6, 7, 7, 8, 9, 10
Step 2: \(n=6\) (even)
Step 3: Median = average of 3rd and 4th values = \(\frac{7 + 8}{2} = 7.5\) days
Answer: 7.5 days

Example 3:
Quantities ordered: 50, 60, 55, 65, 70. Find the median.
Given data: 50, 60, 55, 65, 70
Step 1: Order ascending: 50, 55, 60, 65, 70
Step 2: \(n=5\) (odd)
Step 3: Median = middle value at position 3 = 60 units
Answer: 60 units

Example 4:
Monthly expenditures (Ksh thousands): 150, 165, 155, 160, 170, 175. Find the median.
Given data: 150, 165, 155, 160, 170, 175
Step 1: Order ascending: 150, 155, 160, 165, 170, 175
Step 2: \(n=6\) (even)
Step 3: Median = average of 3rd and 4th values = \(\frac{160 + 165}{2} = 162.5\) Ksh thousands
Answer: 162.5 Ksh thousands

Example 5:
Quantities supplied: 200, 180, 220, 210, 190, 200, 205. Find the median.
Given data: 200, 180, 220, 210, 190, 200, 205
Step 1: Order ascending: 180, 190, 200, 200, 205, 210, 220
Step 2: \(n=7\) (odd)
Step 3: Median = middle value at position 4 = 200 units
Answer: 200 units

Chapter Summary

This chapter covered the fundamental concepts of descriptive statistics essential for data analysis. It began with measures of central tendency, explaining how the mean provides an average value, the mode identifies the most frequently occurring data point, and the median represents the middle value in an ordered data set. The discussion then moved to measures of dispersion, focusing on variance as a measure of data spread around the mean and standard deviation as the square root of variance, which quantifies variability in the same units as the data. Finally, the chapter examined measures of skewness and kurtosis, which describe the asymmetry and peakedness of data distributions respectively, offering deeper insights into the shape and nature of data beyond central tendency and dispersion. These statistical tools collectively enable effective summarization and interpretation of data sets in various practical contexts.