Summarising and analysing data - Mean, median, mode; varianc...

Learning Outcomes

After reading this article, you will be able to: identify and calculate the three main measures of average—mean, median, and mode—for both simple and grouped data; calculate measures of dispersion including variance and standard deviation; understand when each measure is appropriate; interpret what these statistics reveal about a data set; and apply these calculations and concepts to scenarios as required for ACCA exams.

ACCA Management Accounting (MA) Syllabus

For ACCA Management Accounting (MA), you are expected to understand and apply techniques to summarise and analyse data sets. Revision should focus on being able to:

• Calculate and interpret the mean, median, and mode for ungrouped and grouped data
• Calculate variance and standard deviation for both ungrouped and grouped data sets
• Explain the purpose and appropriate use of each statistic
• Apply measures of spread to interpret the consistency and variability of data
• Solve exam questions requiring manual or calculator-based computations

Test Your Knowledge

Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision.

1. A set of daily sales figures (in $): 210, 195, 200, 180, 210, 195, 190. What is the mode?

2. Which measure—mean or median—is less affected by extreme values? Briefly explain why.

3. Given the frequencies below, calculate the mean:

   Value (x)	Frequency (f)
   2	4
   3	7
   4	3
   5	6

4. If a data set has a standard deviation of zero, what does this tell you about the data?

Introduction

Numerical data is often summarised with key statistics so it can be quickly understood. Management accountants frequently need to compare, present, and interpret data using averages and measures of spread. Understanding how to calculate and interpret the mean, median, mode, variance, and standard deviation is therefore a frequent requirement in the ACCA exam.

This article explains each concept, provides step-by-step calculation methods, and shows how to analyse data efficiently and accurately.

MEASURES OF AVERAGE

Averages provide a single value to represent a data set. The three most common are:

• Arithmetic mean
• Median
• Mode

Arithmetic Mean

The mean is the most widely used average and considers every value in the data set. For ungrouped data, add up all values and divide by the number of observations.

Key Term: Arithmetic mean
The sum of all values in the data set divided by the number of values.

For grouped data (where frequencies for each value are known), use:

$\bar{x} = \frac{\sum fx}{\sum f}$

where:

• $f$ = frequency
• $x$ = value

Worked Example 1.1

A company tracks the number of calls received per hour over five hours: 12, 9, 13, 11, 15. What is the mean number of calls?

Answer:
Mean = (12 + 9 + 13 + 11 + 15) ÷ 5 = 60 ÷ 5 = 12 calls per hour.

Median

The median is the middle value when data is arranged in order. For an odd number of values, it is the central item. For an even number, calculate the average of the two central items. The median is not influenced by outliers.

Key Term: Median
The middle value in an ordered data set; or the mean of the two middle values if the set has an even number of entries.

Worked Example 1.2

Find the median of the following set: 18, 12, 20, 15, 17.

Answer:
Ordered set: 12, 15, 17, 18, 20
Median = 17 (the third value out of five).

Mode

The mode is the value that appears most often. Datasets can have more than one mode, or none if all values are unique.

Key Term: Mode
The value that occurs most frequently in a data set.

Worked Example 1.3

The scores from seven tests are: 6, 8, 7, 6, 9, 6, 7. What is the mode?

Answer:
The mode is 6 (appears 3 times).

Comparing Averages

• The mean is affected by extreme (high or low) values.
• The median is less sensitive to outliers, making it better when data is skewed.
• The mode is useful for categorical data or identifying the most popular category.

Revision Tip When data is skewed or has outliers, the median usually gives a better 'typical' value than the mean.

MEASURES OF SPREAD (DISPERSION)

Dispersion shows how spread out data values are from the average. The two most used measures are variance and standard deviation.

Standard Deviation

Standard deviation is a measure of how much individual data points differ from the mean.

Key Term: Standard deviation
The square root of the variance; quantifies the average distance of each data point from the mean.

For a set of $n$ values:

$\sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{n}}$

For data with frequencies:

$\sigma = \sqrt{\frac{\sum fx^2}{\sum f} - \left(\frac{\sum fx}{\sum f}\right)^2}$

Variance

The variance is the mean of the squared differences from the mean. It is in squared units and is less commonly quoted directly than standard deviation.

Key Term: Variance
The average of the squared differences of each data point from the mean.

Worked Example 1.4

Five workers reported the following errors last week: 2, 3, 2, 5, 3.

Calculate:

a) The mean
b) The variance
c) The standard deviation

Answer:
a) Mean = (2 + 3 + 2 + 5 + 3)/5 = 15/5 = 3
b) Find squared differences: (2-3)²=1, (3-3)²=0, (2-3)²=1, (5-3)²=4, (3-3)²=0
Sum = 1+0+1+4+0=6
Variance = 6/5 = 1.2
c) Standard deviation = √1.2 ≈ 1.10

Coefficient of Variation

The coefficient of variation (CV) allows for comparison of variability between data sets with different means. It is calculated as:

$CV = \frac{\text{Standard deviation}}{\text{Mean}}$

A higher CV indicates more variability relative to the mean.

Worked Example 1.5

Company X has mean sales of $80,000 with a standard deviation of $16,000. Company Y has mean sales of $150,000 with a standard deviation of $18,000.

Which company has more relative sales variability?

Answer:
X: CV = 16,000/80,000 = 0.2 (20%)
Y: CV = 18,000/150,000 = 0.12 (12%)
X has more variability relative to its mean.

APPLICATION TO GROUPED DATA

When data is presented in classes or intervals, use the class midpoint as the value for calculations.

Worked Example 1.6

The following table shows employee overtime hours in a week:

Calculate the mean overtime hours.

Answer:
$\bar{x} = \frac{(5×6)+(10×12)+(15×7)+(20×5)}{6+12+7+5} = \frac{30+120+105+100}{30} = \frac{355}{30} ≈ 11.83 \text{ hours}$

INTERPRETING DATA USING AVERAGES AND SPREAD

• Mean and standard deviation are best for data with few outliers.
• Median and interquartile range are preferable for highly skewed data.
• Mode is best for non-numeric or categorical data.

Exam Warning

Take care when data is presented in a frequency table or grouped intervals. Use class midpoints for all calculations and ensure that the total frequency is correct. Failing to do so may lead to computational errors, and exam questions often test accuracy in these steps.

Summary

Mean, median, and mode provide different ways to summarise data with a single value. Each has its own advantages depending on the data type and distribution. Variance and standard deviation quantify how widely values in a dataset are dispersed, which is essential for comparing risk, consistency, or process stability. Know when and how to use each calculation and always apply the formula appropriate to the data format given.

Key Point Checklist

This article has covered the following key knowledge points:

• Calculate the mean, median, and mode for ungrouped and grouped data
• Identify when each average is most appropriate and interpret results
• Calculate variance and standard deviation—understand their meaning
• Use the coefficient of variation for comparing variability across data sets
• Apply the correct method to data in class intervals or frequency tables

Key Terms and Concepts

• Arithmetic mean
• Median
• Mode
• Standard deviation
• Variance