Compiled by: Grace Nganga
Descriptive statistics is essential in professional settings across Kenya, enabling decision-makers in hospitals, banks, universities, and county government offices to summarize and interpret data effectively. This chapter introduces measures of central tendency which help to identify typical values in data sets, facilitating informed planning, resource allocation, and performance evaluation. Understanding mean, mode, and median supports professionals in diverse sectors such as agriculture, retail, and healthcare to analyze trends and make evidence-based decisions.
Measures of central tendency are statistical values that represent a typical or central point in a data set. They simplify large volumes of data by providing a single value that summarizes the data’s general pattern. In Kenya’s professional environments like SACCOs, county hospitals, and educational institutions, these measures help in analyzing customer transactions, patient wait times, and student scores respectively, to inform management strategies and improve service delivery.
The mean or average is the sum of all data values divided by the number of values. It provides a measure of the central location of the data and is widely used in sectors such as retail business to analyze average sales, or in county government offices to calculate average employee salaries.
$$\text{Mean} = \frac{\sum x_i}{n}$$
where \(x_i\) are data points and \(n\) is the number of data points.
Example 1: A retail store records daily sales (in Ksh) over 5 days as follows: 12,000; 15,000; 14,000; 13,000; 16,000. Find the mean sales.
Given: Sales = 12,000; 15,000; 14,000; 13,000; 16,000
\(n = 5\)
$$\text{Mean} = \frac{12,000 + 15,000 + 14,000 + 13,000 + 16,000}{5}$$$$= \frac{70,000}{5}$$$$= 14,000 \text{ Ksh}$$ Answer: 14,000 Ksh
Example 2: A county referral hospital records the number of patients served in a week: 120, 135, 110, 145, 130, 125, 140. Calculate the mean number of patients served per day.
Given: Patients = 120, 135, 110, 145, 130, 125, 140
\(n = 7\)
$$\text{Mean} = \frac{120 + 135 + 110 + 145 + 130 + 125 + 140}{7}$$$$= \frac{905}{7}$$$$= 129.29 \text{ patients}$$ Answer: 129.29 patients
Example 3: A SACCO records loan amounts disbursed (in Ksh) for 8 clients: 50,000; 60,000; 55,000; 65,000; 70,000; 58,000; 62,000; 68,000. Find the mean loan amount.
Given: Loans = 50,000; 60,000; 55,000; 65,000; 70,000; 58,000; 62,000; 68,000
\(n = 8\)
$$\text{Mean} = \frac{50,000 + 60,000 + 55,000 + 65,000 + 70,000 + 58,000 + 62,000 + 68,000}{8}$$$$= \frac{488,000}{8}$$$$= 61,000 \text{ Ksh}$$ Answer: 61,000 Ksh
Example 4: A university records the scores of 10 students in a statistics exam: 45, 60, 55, 70, 65, 80, 75, 68, 62, 58. Calculate the mean score.
Given: Scores = 45, 60, 55, 70, 65, 80, 75, 68, 62, 58
\(n = 10\)
$$\text{Mean} = \frac{45 + 60 + 55 + 70 + 65 + 80 + 75 + 68 + 62 + 58}{10}$$$$= \frac{638}{10}$$$$= 63.8 \text{ marks}$$ Answer: 63.8 marks
Example 5: A farm harvests the following amounts of maize (in bags) over 6 weeks: 120, 135, 130, 125, 140, 128. Find the average weekly harvest.
Given: Harvest = 120, 135, 130, 125, 140, 128
\(n = 6\)
$$\text{Mean} = \frac{120 + 135 + 130 + 125 + 140 + 128}{6}$$$$= \frac{778}{6}$$$$= 129.67 \text{ bags}$$ Answer: 129.67 bags
The mode is the value that appears most frequently in a data set. It is useful for identifying the most common category or value, such as the most frequent product sold in a retail store or the most common patient diagnosis in a hospital. Mode can be used with both numerical and categorical data.
Example 1: A county hospital records the number of daily emergency cases over a week: 12, 15, 12, 18, 20, 15, 12. Find the mode.
Given: Cases = 12, 15, 12, 18, 20, 15, 12
The value 12 appears 3 times, 15 appears 2 times, others less.
Answer: 12 cases
Example 2: A retail shop logs the number of customers per day over 8 days: 30, 35, 40, 35, 30, 35, 40, 45. Identify the mode.
Given: Customers = 30, 35, 40, 35, 30, 35, 40, 45
35 appears 3 times, 30 and 40 appear twice, 45 once.
Answer: 35 customers
Example 3: A university records the marks of 10 students: 50, 55, 65, 55, 60, 70, 55, 65, 60, 70. Find the mode.
Given: Scores = 50, 55, 65, 55, 60, 70, 55, 65, 60, 70
55 appears 3 times, 65 and 60 appear twice, 70 twice.
Answer: 55 marks
Example 4: A SACCO records loan categories by size (in Ksh): 30,000; 40,000; 30,000; 50,000; 30,000; 40,000; 60,000. Find the mode.
Given: Loans = 30,000; 40,000; 30,000; 50,000; 30,000; 40,000; 60,000
30,000 appears 3 times, 40,000 twice, others once.
Answer: 30,000 Ksh
Example 5: A farm records the number of livestock sold per month over 6 months: 15, 20, 15, 25, 20, 15. Identify the mode.
Given: Sales = 15, 20, 15, 25, 20, 15
15 appears 3 times, 20 twice, 25 once.
Answer: 15 livestock
The median is the middle value of a data set when arranged in ascending or descending order. It divides the data into two halves, with 50% of data points below and 50% above. Median is especially useful when data has outliers, such as in income distribution in a county or patient waiting times at a hospital.
Example 1: Find the median of daily sales (Ksh) for 7 days: 12,000; 15,000; 14,000; 13,000; 16,000; 11,000; 17,000.
Given: Sales = 12,000; 15,000; 14,000; 13,000; 16,000; 11,000; 17,000
Order data: 11,000; 12,000; 13,000; 14,000; 15,000; 16,000; 17,000
\(n=7\) (odd)
Median position = \(\frac{7+1}{2} = 4\)th value
Median = 14,000 Ksh
Answer: 14,000 Ksh
Example 2: Find the median patient count for 6 days: 120, 135, 110, 145, 130, 125.
Given: Counts = 120, 135, 110, 145, 130, 125
Order data: 110, 120, 125, 130, 135, 145
\(n=6\) (even)
Median position = average of 3rd and 4th values: (125 + 130)/2
$$\frac{125 + 130}{2} = \frac{255}{2} = 127.5$$
Answer: 127.5 patients
Example 3: Find the median loan amount (Ksh) for 5 clients: 50,000; 60,000; 55,000; 65,000; 70,000.
Given: Loans = 50,000; 60,000; 55,000; 65,000; 70,000
Order data: 50,000; 55,000; 60,000; 65,000; 70,000
\(n=5\) (odd)
Median position = \(\frac{5+1}{2} = 3\)rd value
Median = 60,000 Ksh
Answer: 60,000 Ksh
Example 4: Find the median student scores: 45, 60, 55, 70, 65, 80, 75, 68.
Given: Scores = 45, 60, 55, 70, 65, 80, 75, 68
Order data: 45, 55, 60, 65, 68, 70, 75, 80
\(n=8\) (even)
Median position = average of 4th and 5th values: (65 + 68)/2
$$\frac{65 + 68}{2} = \frac{133}{2} = 66.5$$
Answer: 66.5 marks
Example 5: Find the median weekly livestock sales: 15, 20, 15, 25, 20, 15, 30.
Given: Sales = 15, 20, 15, 25, 20, 15, 30
Order data: 15, 15, 15, 20, 20, 25, 30
\(n=7\) (odd)
Median position = 4th value
Median = 20 livestock
Answer: 20 livestock
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Create a free accountThis chapter covered key concepts in descriptive statistics essential for summarizing and interpreting data sets effectively. It began with measures of central tendency, explaining how the mean provides the arithmetic average, the mode identifies the most frequently occurring value, and the median represents the middle value when data are ordered. The focus then shifted to measures of dispersion, detailing how variance quantifies the average squared deviation from the mean, while standard deviation offers a measure of spread in the same units as the data. Finally, the chapter introduced measures of skewness and kurtosis, which describe the asymmetry and peakedness of a data distribution respectively, providing deeper insights into the shape and characteristics of the data beyond central tendency and variability. Together, these statistical tools form a foundation for analyzing and understanding data patterns in practical applications.
A county referral hospital recorded the daily number of patients attending its outpatient department over 5 days as follows: 45, 52, 48, 50, and 55. Calculate the mean number of patients per day. (2 marks)
A SACCO has 7 loan officers with the following monthly loan disbursement amounts (in Ksh thousands): 120, 135, 120, 140, 150, 120, 130. Find the mode of the loan disbursements. (2 marks)