Application of Measures of Central Tendency to Analyse Given Information – JSS3 Mathematics Lesson Note (Third Term)

Measures of central tendency are single values that show the centre of a set of data: the mean, the median and the mode.

Behavioural objectives

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

1. Calculate the range of a set of ungrouped data.
2. Calculate the mean of a set of ungrouped data.
3. Find the median of a set of numbers with an odd or an even number of terms.
4. Find the mode of a set of numbers or of a frequency table.
5. Use the mean to solve problems about ages.
6. Choose a suitable measure to describe a given set of data.

What are the measures of central tendency?

There are three measures of central tendency: the mean, the median and the mode.

• Mean: the sum of the numbers divided by how many numbers there are.
• Median: the middle number when the data is arranged in order.
• Mode: the number that appears most often.
• Range: the largest number minus the smallest. It measures how spread out the data is, so it is not a measure of the centre.

What is the range?

The range of a set of numbers is the difference between the largest and the smallest numbers.

Steps to find the range

1. Arrange the data in order of size.
2. Pick the largest and the smallest numbers.
3. Subtract the smallest from the largest.

Example

Find the range of these scores: 79, 50, 52, 34, 68, 60.

1. Rank order: 79, 68, 60, 52, 50, 34.
2. Largest = 79 and smallest = 34.
3. Range = 79 minus 34 = 45.

What is the mean?

The mean, or arithmetic mean, is the sum of all the numbers in a set divided by the number of numbers.

• Formula: Mean = (sum of the numbers in the set) ÷ n.
• n: the number of items in the set.
• Other name: the mean is the most common kind of average.

Steps to find the mean

1. Add all the numbers.
2. Count how many numbers there are (n).
3. Divide the sum by n.

Example 1: Mean of a set of numbers

Calculate the mean of: 176, 174, 178, 181, 174, 175, 179, 107, 177, 102.

1. Sum = 176 + 174 + 178 + 181 + 174 + 175 + 179 + 107 + 177 + 102 = 1623.
2. There are 10 numbers, so n = 10.
3. Mean = 1623 ÷ 10 = 162.3.

Example 2: Mean and ages

Five children have an average age of 7 years 11 months. Without the youngest child, the average is 8 years 4 months. Find the age of the youngest child.

1. Total age of all five children = 5 × 7 y 11 mo = 35 y 55 mo = 39 y 7 mo.
2. Total age of the four older children = 4 × 8 y 4 mo = 32 y 16 mo = 33 y 4 mo.
3. Age of the youngest = 39 y 7 mo minus 33 y 4 mo = 6 y 3 mo.

• Tip: mean × number of items gives the total.
• Tip: 12 months make 1 year, so change extra months to years.

What is the median?

The median is the middle term of a set of numbers arranged in order of size.

• Odd number of terms: the median is the single middle term.
• Even number of terms: the median is the mean of the two middle terms.

Steps to find the median

1. Arrange the numbers in order, from lowest to highest or highest to lowest.
2. Count the terms.
3. Pick the middle term, or add the two middle terms and divide by 2.

Example 1: Odd number of terms

Find the median of 15, 11, 8, 21, 17.

1. Rank order: 21, 17, 15, 11, 8.
2. There are five numbers, so the median is the 3rd number.
3. Median = 15.

Example 2: Even number of terms

Find the median of 38, 21, 14, 4, 43, 9, 2, 50.

1. Order: 2, 4, 9, 14, 21, 38, 43, 50.
2. There are eight numbers, so the middle terms are the 4th and 5th: 14 and 21.
3. Median = (14 + 21) ÷ 2 = 17.5.

What is the mode?

The mode is the number or value that appears most often, that is, the value with the greatest frequency.

• One mode: one value has the highest frequency, as in 2, 3, 3, 5 (mode 3).
• More than one mode: two or more values tie for the highest frequency, as in 1, 1, 2, 3, 3 (modes 1 and 3).
• No mode: every value appears once, as in 4, 6, 8, 9.

Example: Mode and median from a frequency table

Twenty-one students measured the melting point of naphthalene. The table shows the results. Find the modal temperature and the median temperature.

Modal temperature

1. Find the highest frequency: 7.
2. Seven students recorded 80°C.
3. Mode = 80°C.

Median temperature

1. There are 21 students, so the median is the 11th value.
2. Add the frequencies in order: 1, then 1 + 2 = 3, then 3 + 7 = 10.
3. The first 10 values end at 80°C, so the 11th value is the first of the 81°C results.
4. Median = 81°C.

How do we choose the best measure to describe data?

Each measure answers a different question, so the best one depends on the data and on what we want to know.

Take the data 2, 3, 3, 5, 7. The mean is 20 ÷ 5 = 4, the median is 3, the mode is 3 and the range is 7 minus 2 = 5.

Worked application

Look again at the data in Example 1 under the mean.

• Mean: 162.3, which is lower than 8 of the 10 values.
• Reason: the two small values, 107 and 102, pull the mean down.
• Median: the ordered middle terms are 175 and 176, so the median is 175.5.
• Mode: 174, which appears twice.
• Conclusion: the median describes the typical value better than the mean here.

Evaluation

1. Find the mean of the numbers 13, 2, 8, 5 and 11. Also find the range.
2. A mother has seven children. The mean age of the children is 13 years 2 months. With the mother’s age included, the mean age rises to 17 years 7 months. Calculate the age of the mother.
3. For the set 12, 14, 14, 15, 18, 18, 19, 19, 21:
   1. State the median.
   2. State the mode or modes.
   3. Calculate the mean, correct to 1 decimal place.
4. Explain why the median describes the data better than the mean for the numbers 3, 4, 4, 5, 90.

Assignment

The number of goals scored by a team in nine handball matches are: 3, 5, 7, 7, 8, 4, 1, 1, 5.

1. Find the mean, correct to 1 decimal place, the median and the range.
2. State every mode.
3. Say which of these statements are true: (i) the mean is greater than the mode; (ii) the mode and the median are equal; (iii) the mean, median and mode are all equal. Give a reason for each answer.