Representing and Interpreting Data for JSS 1

Data representation

Performance Objectives

By the end of the lesson, students should be able to:

1. Determine the median of a given data set.

Meaning of Data and Data Set

Data is a collection of facts, numbers, measurements or observations gathered for a specific purpose.

A data set is a complete collection of these related pieces of information grouped together for analysis.

For example, the scores of ten students in a mathematics test form a data set. In everyday statistics, data must be organized and represented clearly so that meaningful conclusions can be drawn from it.

Meaning of Median

The median is the middle value in a set of data when the numbers are arranged in order of size, either from the smallest to the largest or from the largest to the smallest.

The median divides a data set into two equal halves. One half of the data values are less than or equal to the median, while the other half of the data values are greater than or equal to the median.

The Importance of Ordering Data

Before finding the median of any data set, the data must first be arranged in order of size. This process is called sorting or ordering the data. Arranging data from the smallest value to the largest value is called ascending order, while arranging from the largest to the smallest is called descending order.

If the data is not ordered, the physical middle number in the raw list will not represent the true mathematical middle value. Ordering the data is a logical step that ensures every value is weighed correctly according to its actual size.

Determining the Median of an Odd Number Data Set

When a data set contains an odd number of values, there is exactly one middle value. The median is simply that single middle number after the data has been arranged in order of size.

To find the position of the median in an odd data set, the following formula can be used:

$$Position = \frac{n + 1}{2}$$

In this formula, $n$ represents the total number of values in the data set.

The step-by-step procedure for finding the median of an odd number data set involves the following steps:

1. Count the total number of values ($n$) to confirm that the count is odd.
2. Arrange the data values in ascending order (from smallest to largest).
3. Find the middle value either by counting from both ends to the center or by using the position formula.
4. Identify the value at that middle position as the median.

The diagram below illustrates how to identify the median in an odd number data set.

Worked Example 1

Five JSS 1 students scored the following marks in a weekly mathematics quiz: $12, 15, 10, 18, \text{ and } 14$. Determine the median score.

Solution:

First, count the number of values in the data set:

$$n = 5$$

Since $5$ is an odd number, there will be exactly one middle value.

Next, arrange the scores in ascending order:

$$10, 12, 14, 15, 18$$

Determine the position of the median:

$$Position = \frac{5 + 1}{2} = \frac{6}{2} = 3\text{-rd position}$$

Identify the value at the 3rd position in the ordered list:

The 1st value is $10$.

The 2nd value is $12$.

The 3rd value is $14$.

Therefore, the median score is $14$.

Worked Example 2

The daily afternoon temperatures in degrees Celsius recorded in Enugu over a week were: $32, 29, 35, 30, 33, 31, \text{ and } 34$. Find the median temperature.

Solution:

First, count the number of values in the data set:

$$n = 7$$

Since $7$ is an odd number, there is a single middle value.

Next, arrange the temperatures in ascending order:

$$29, 30, 31, 32, 33, 34, 35$$

Determine the position of the median:

$$Position = \frac{7 + 1}{2} = \frac{8}{2} = 4\text{-th position}$$

Identify the value at the 4th position in the ordered list:

The 4th value is $32$.

Therefore, the median temperature is $32^\circ\text{C}$.

Worked Example 3

A school agricultural farm harvested nine tubers of yam with the following weights in kilograms: $4, 2, 7, 5, 9, 3, 6, 8, \text{ and } 5$. Determine the median weight of the harvested yams.

Solution:

First, count the number of values in the data set:

$$n = 9$$

Since $9$ is an odd number, there is a single middle value.

Next, arrange the weights in ascending order:

$$2, 3, 4, 5, 5, 6, 7, 8, 9$$

Determine the position of the median:

$$Position = \frac{9 + 1}{2} = \frac{10}{2} = 5\text{-th position}$$

Identify the value at the 5th position in the ordered list:

The 5th value is $5$.

Therefore, the median weight of the yams is $5\text{ kg}$.

Determining the Median of an Even Number Data Set

When a data set contains an even number of values, there is no single middle value. Instead, there are two middle values. The median is determined by finding the average (mean) of these two middle values.

To find the two middle positions in an even data set, the following formulas are used:

$$First\text{ }Middle\text{ }Position = \frac{n}{2}$$
$$Second\text{ }Middle\text{ }Position = \frac{n}{2} + 1$$

In these formulas, $n$ represents the total number of values in the data set.

The step-by-step procedure for finding the median of an even number data set involves the following steps:

1. Count the total number of values ($n$) to confirm that the count is even.
2. Arrange the data values in ascending order (from smallest to largest).
3. Identify the two middle values in the ordered list.
4. Add the two middle values together.
5. Divide the sum by $2$ to obtain the median.

The diagram below illustrates how to identify and calculate the median in an even number data set.

Worked Example 1

The ages of six children in a household are: $14, 10, 16, 12, 18, \text{ and } 20$ years. Determine the median age.

Solution:

First, count the number of values in the data set:

$$n = 6$$

Since $6$ is an even number, there are two middle values.

Next, arrange the ages in ascending order:

$$10, 12, 14, 16, 18, 20$$

Determine the positions of the two middle values:

$$First\text{ }Middle\text{ }Position = \frac{6}{2} = 3\text{-rd position}$$
$$Second\text{ }Middle\text{ }Position = \frac{6}{2} + 1 = 4\text{-th position}$$

Identify the values at the 3rd and 4th positions in the ordered list:

The 3rd value is $14$.

The 4th value is $16$.

Calculate the median by finding the average of these two values:

$$Median = \frac{14 + 16}{2}$$
$$Median = \frac{30}{2}$$
$$Median = 15$$

Therefore, the median age is $15$ years.

Worked Example 2

Eight bags of cocoa beans have the following weights in kilograms: $45, 50, 42, 48, 52, 47, 46, \text{ and } 51$. Find the median weight of the cocoa bags.

Solution:

First, count the number of values in the data set:

$$n = 8$$

Since $8$ is an even number, there are two middle values.

Next, arrange the weights in ascending order:

$$42, 45, 46, 47, 48, 50, 51, 52$$

Determine the positions of the two middle values:

$$First\text{ }Middle\text{ }Position = \frac{8}{2} = 4\text{-th position}$$
$$Second\text{ }Middle\text{ }Position = \frac{8}{2} + 1 = 5\text{-th position}$$

Identify the values at the 4th and 5th positions in the ordered list:

The 4th value is $47$.

The 5th value is $48$.

Calculate the median by finding the average of these two values:

$$Median = \frac{47 + 48}{2}$$
$$Median = \frac{95}{2}$$
$$Median = 47.5$$

Therefore, the median weight of the cocoa bags is $47.5\text{ kg}$.

Worked Example 3

The number of goals scored by a local football club in ten consecutive matches are: $2, 1, 0, 3, 2, 1, 4, 0, 2, \text{ and } 5$. Determine the median number of goals scored.

Solution:

First, count the number of values in the data set:

$$n = 10$$

Since $10$ is an even number, there are two middle values.

Next, arrange the goals in ascending order:

$$0, 0, 1, 1, 2, 2, 2, 3, 4, 5$$

Determine the positions of the two middle values:

$$First\text{ }Middle\text{ }Position = \frac{10}{2} = 5\text{-th position}$$
$$Second\text{ }Middle\text{ }Position = \frac{10}{2} + 1 = 6\text{-th position}$$

Identify the values at the 5th and 6th positions in the ordered list:

The 5th value is $2$.

The 6th value is $2$.

Calculate the median by finding the average of these two values:

$$Median = \frac{2 + 2}{2}$$
$$Median = \frac{4}{2}$$
$$Median = 2$$

Therefore, the median number of goals scored is $2$.

Logical Thinking: Comparing Median with Raw Data

The median is a highly reliable measure of central tendency because it is not easily affected by extremely high or extremely low values, which are known as outliers. This makes the median a very useful tool when analyzing real-world statistics, such as household incomes or test scores, where a single unusual value could distort other averages like the mean.

The table below compares how the median behaves in different data sets to show its stability.

As shown in the table, even when the highest value in the data set changes dramatically (from $18$ to $90$ in the odd set, or from $12$ to $100$ in the even set), the median remains exactly the same. This demonstrates the logical value of the median in representing the true center of a data set without being distorted by extreme values.

Summary of Key Steps

To find the median of any given data set, the following systematic approach should always be followed:

1. Arrange the raw data in ascending order.
2. Count the total number of data points, $n$.
3. If $n$ is odd, select the single middle value at position $\frac{n+1}{2}$.
4. If $n$ is even, locate the two middle values at positions $\frac{n}{2}$ and $\frac{n}{2} + 1$, add them together, and divide by $2$.

Class Activity and Discussion

To practice logical thinking and weighing evidence, consider the following scenario:

A small business has five employees who earn the following monthly salaries: $\text{₦}30,000$, $\text{₦}32,000$, $\text{₦}35,000$, $\text{₦}38,000$, and $\text{₦}250,000$ (the manager’s salary). If someone asks for the typical salary of a worker at this business, would the median salary or the average (mean) salary give a fairer picture? Discuss this with classmates by calculating both values and comparing them.

Exercises

Section A: Multiple Choice Questions

Choose the correct option for each of the following questions:

1. What is the first step in finding the median of a given data set?
   1. Add all the numbers together
   2. Divide the total number of values by two
   3. Arrange the numbers in order of size
   4. Subtract the smallest number from the largest number
2. Find the median of the following data set: $7, 3, 9, 5, 11$.
   1. $9$
   2. $7$
   3. $5$
   4. $3$
3. What is the median of the numbers: $12, 8, 15, 10, 18, 14$?
   1. $12$
   2. $13$
   3. $14$
   4. $15$
4. If a data set has $15$ values arranged in ascending order, which position holds the median?
   1. $7\text{-th position}$
   2. $8\text{-th position}$
   3. $9\text{-th position}$
   4. $10\text{-th position}$
5. A data set consists of the values: $4, 9, 15, 2, 8, 12, 1, 20$. What is the median?
   1. $8$
   2. $8.5$
   3. $9$
   4. $10$

Section B: Theory and Calculations

Solve the following problems completely, showing all steps clearly:

1. The heights in centimeters of seven JSS 1 students were measured as follows: $135, 142, 130, 148, 138, 140, \text{ and } 145$.
   1. Arrange the heights in ascending order.
   2. Determine the median height of the students.
2. A poultry farmer recorded the number of eggs collected daily for ten days: $24, 28, 22, 26, 30, 25, 27, 29, 23, \text{ and } 31$.
   1. Arrange the data in order of size.
   2. Identify the two middle values.
   3. Calculate the median number of eggs collected.
3. The scores of nine students in a civic education test are: $8, 5, 7, 6, 9, 4, 8, 7, \text{ and } 10$. Find the median score.
4. A trader recorded the daily sales of bags of rice over six days: $12, 15, 9, 11, 14, \text{ and } 10$. Determine the median number of bags sold.