Descriptive Statistics

Introduction

Descriptive statistics provides the methods used to organise, summarise and communicate the main features of a dataset. In business analytics, managers rarely begin with a raw spreadsheet and immediately build a predictive model. They first need to understand what the data contain: how often events occur, what a typical observation looks like, how much observations differ, whether the distribution is symmetric, and whether two variables move together.

Descriptive statistics does not establish causation and does not generalise beyond the data without additional inferential methods. Its purpose is disciplined description. A good descriptive analysis converts a large collection of observations into information that supports operational, financial and managerial decisions.

Examples include summarising monthly sales, comparing customer waiting times, describing employee salaries, identifying unusually large transactions and assessing whether advertising expenditure tends to vary with revenue.

Learning objectives

By the end of this chapter, you should be able to:

1. Construct and interpret frequency distributions.
2. Calculate and explain measures of central tendency.
3. Calculate and interpret measures of dispersion and position.
4. Describe skewness and kurtosis and recognise their business implications.
5. Calculate covariance and distinguish it from correlation.
6. Calculate and interpret standardised scores.
7. Select appropriate descriptive measures while recognising their assumptions and limitations.

1. Data, variables and distributions

An observation is one recorded unit, such as one customer, invoice or day. A variable is a characteristic measured for each observation, such as purchase value, region or delivery time. A dataset may contain:

• Categorical variables, whose values represent groups or labels, such as product category or payment method.
• Numerical variables, whose values represent counts or measurements. Discrete variables are countable, such as number of complaints; continuous variables can take values on an interval, such as delivery time.
• Ordinal variables, whose categories have an order but not necessarily equal spacing, such as satisfaction ratings from very dissatisfied to very satisfied.

The measurement scale affects which summaries are meaningful. Frequencies and proportions can be used for categorical data. Means require numerical data for which addition and equal intervals are meaningful. A median is often suitable for ordinal or skewed numerical data.

A frequency distribution lists observed values or intervals together with their frequencies. If value xᵢ occurs fᵢ times in a dataset of n observations, then:

The relative frequency is:

where fᵢ is the frequency of category or class i, n is the total number of observations, and pᵢ is the proportion. Percentage frequency is 100pᵢ. The relative frequencies should sum to 1, apart from rounding.

Grouped frequency distributions

When there are many numerical values, values can be grouped into class intervals. A useful table includes the class limits, frequency, relative frequency and cumulative frequency. Class intervals should be mutually exclusive and collectively exhaustive: every observation belongs to one and only one class.

For continuous measurements, class boundaries may be used. For example, intervals 0–9 and 10–19 can be interpreted as boundaries 0 ≤ x < 10 and 10 ≤ x < 20. The choice of interval width affects the appearance of a histogram, so analysts should state the rule used.

A histogram displays frequencies for numerical intervals, with adjacent bars. A bar chart displays categories and normally has separated bars. Confusing these graphs can produce misleading interpretations.

2. Measures of central tendency

Central tendency describes a typical or central value. The mean, median and mode answer different questions and should not be treated as interchangeable.

Arithmetic mean

For n observations x₁, x₂, …, xₙ, the sample arithmetic mean is:

The mean uses every observation and is appropriate for numerical data when extreme values do not distort the summary excessively. It is sensitive to outliers. For frequency data, the weighted mean is:

Here xᵢ is a value or class midpoint and fᵢ is its frequency. When grouped data are used, class midpoints approximate the original observations, so the result is an estimate rather than an exact mean.

Median

The median is the middle observation after sorting the data. If n is odd, it is the observation at position (n + 1)/2. If n is even, it is the average of the observations at positions n/2 and n/2 + 1. The median is resistant to extreme values and is often preferred for income, house prices and transaction values.

Mode

The mode is the most frequently occurring value or category. A dataset may be unimodal, bimodal or have no unique mode. The mode is especially useful for categorical data, where a mean may have no meaningful interpretation.

Worked example 1: central tendency

Consider daily online orders over seven days: 18, 21, 21, 24, 26, 27 and 63.

The mean is:

The median is the fourth ordered value, 24. The mode is 21. The unusually high value of 63 pulls the mean above the median. If the question is “what was a typical day?”, the median may be more representative. If the question concerns total orders allocated across seven equal days, the mean remains useful.

3. Measures of dispersion

Two datasets can have the same mean but very different levels of risk or inconsistency. Dispersion measures quantify variation around the centre.

Range and interquartile range

The range is:

It is simple but depends only on two observations. The interquartile range is:

where Q₁ is the first quartile and Q₃ is the third quartile. The IQR covers the middle 50% of observations and is resistant to extreme values. Quartile conventions differ slightly between software packages; the selected convention should be stated for small datasets.

Variance and standard deviation

For a sample, the sample variance is:

The sample standard deviation is:

The denominator n − 1, called Bessel’s correction, is used when the sample variance estimates the variance of a wider population. If the dataset is the complete population of interest, the population variance is:

and the population standard deviation is σ = √σ². Here N is population size and μ is the population mean. Variance is expressed in squared units; standard deviation has the same units as the original variable.

The standard deviation is appropriate when deviations from the mean are meaningful and the mean is a suitable centre. It is sensitive to outliers.

Worked example 2: sample variance and standard deviation

Suppose four delivery times, in minutes, are 8, 10, 12 and 14. The sample mean is 11. Deviations from the mean are −3, −1, 1 and 3; squared deviations are 9, 1, 1 and 9. Therefore:

The standard deviation indicates the typical scale of variation around the sample mean, but it is not itself the average absolute deviation.

Coefficient of variation

For a positive variable measured on a ratio scale, the coefficient of variation is:

It compares relative variability across datasets with different units or means. It should not be used when the mean is zero or close to zero, or for variables whose zero point is arbitrary, such as temperature in Celsius.

4. Measures of position

Measures of position locate an observation within an ordered dataset.

A percentile is a value below which a specified percentage of observations fall. The 50th percentile is the median; the 25th and 75th percentiles correspond to Q₁ and Q₃. Percentiles are useful in service-level monitoring, examination results and salary benchmarking.

A simple percentile rank can be expressed as:

Different statistical systems use different interpolation rules, especially when x is not an observed value. Therefore, percentile results from software should be interpreted with awareness of the selected method.

Standardised scores

A z-score expresses an observation in standard deviation units from the mean:

For a sample-based standardisation, x̄ and s are often substituted:

The numerator and denominator have the same units, so z is unit-free. A positive z-score is above the reference mean; a negative score is below it. A z-score of 2 means the observation is two reference standard deviations above the mean. A large absolute z-score can flag a potential outlier, but it does not prove that the observation is erroneous.

Worked example 3: comparing different scales

An employee processed 42 claims when the team mean was 30 and the standard deviation was 6:

Another employee achieved a customer rating of 8.4 when the mean was 7.5 and the standard deviation was 0.3:

Relative to their respective teams, the second performance is farther above its mean, although the original units are different.

5. Shape: skewness and kurtosis

Distribution shape adds information beyond centre and spread.

Skewness

A symmetric distribution has similar tails on both sides of its centre. A positively skewed distribution has a long or heavy right tail; a negatively skewed distribution has a long or heavy left tail. Business variables such as personal income, claim size and customer spending are often positively skewed because a small number of observations are very large.

A commonly used moment coefficient of skewness is:

where m₂ = (1/n)∑(xᵢ − x̄)² and m₃ = (1/n)∑(xᵢ − x̄)³. This is a population-style descriptive moment coefficient. Software may instead report an adjusted sample coefficient, so the formula and convention matter. Skewness has no units.

For a positively skewed distribution, the mean is often greater than the median; for a negatively skewed distribution, the mean is often less than the median. This is a useful tendency, not a universal law.

Kurtosis

Kurtosis describes tail heaviness and concentration relative to a reference distribution. The fourth standardized moment is:

where m₄ = (1/n)∑(xᵢ − x̄)⁴. Excess kurtosis is:

Under this moment convention, a normal distribution has kurtosis 3 and excess kurtosis 0. Positive excess kurtosis indicates relatively heavy tails and greater potential for extreme observations; negative excess kurtosis indicates relatively light tails. Because estimators and software definitions differ, kurtosis should not be interpreted without checking whether the output is kurtosis or excess kurtosis and whether it is bias-adjusted.

Skewness and kurtosis are descriptive indicators, not automatic tests of normality. Histograms, boxplots, domain knowledge and formal tests may be used together.

6. Covariance and association

Covariance describes the direction in which two numerical variables vary together. For paired sample observations (xᵢ, yᵢ), sample covariance is:

A positive covariance indicates that above-average values of x tend to occur with above-average values of y. A negative covariance indicates an opposite tendency. A covariance near zero indicates little linear co-movement, although nonlinear association may still exist.

Covariance depends on measurement units, making its magnitude difficult to compare across variable pairs. Pearson’s correlation standardises covariance:

Provided both standard deviations are nonzero, r lies between −1 and +1. Correlation measures linear association, not causation. A strong correlation may result from a third variable, common trends or data-processing artefacts.

Worked example 4: covariance

Advertising expenditure x and sales y for four weeks are:

The means are x̄ = 2.5 and ȳ = 3.5. The products of deviations are 1.5, 0.25, 0.75 and 0.5, which sum to 3. Thus:

The positive covariance indicates positive co-movement in these observations. However, four weeks are insufficient to establish that advertising causes sales.

7. An integrated business case

Background

The following is an illustrative case. A regional delivery company wants to assess whether its new service process is producing consistent delivery times and whether high delivery times are concentrated among particular orders.

Problem

The operations manager has 10 delivery times, in minutes:

42, 45, 47, 48, 50, 51, 53, 55, 58, 91.

The manager must report a typical delivery time, variation, possible unusual observations and implications for service improvement.

Analysis

The sum is 540, so the mean is 54 minutes. The median is the average of the fifth and sixth values: (50 + 51)/2 = 50.5 minutes. The range is 91 − 42 = 49 minutes.

The lower half is 42, 45, 47, 48, 50 and the upper half is 51, 53, 55, 58, 91. Using the median-of-halves convention, Q₁ = 47 and Q₃ = 55, so IQR = 8 minutes. The upper outlier fence is Q₃ + 1.5IQR = 55 + 12 = 67 minutes. The 91-minute delivery is therefore a potential outlier under the boxplot rule.

For the sample, the deviations from the mean are −12, −9, −7, −6, −4, −3, −1, 1, 4 and 37. Their squared deviations sum to 1,800. Hence:

The mean of 54 minutes and standard deviation of approximately 14.14 minutes are strongly affected by the 91-minute delivery. Without that observation, the mean is 49.89 minutes and the sample standard deviation is approximately 5.39 minutes. This comparison does not justify deleting the observation; it shows why its operational cause must be investigated.

Approach and solution

The company should first verify whether 91 minutes is a data-entry error, a delayed route, a customer absence or a genuine service failure. If genuine, it should remain in the operational report, because extreme service failures matter to customers. Management may report both the mean and median, together with the IQR and a separately investigated exception count.

A dashboard could display the median delivery time, the 90th or 95th percentile, the proportion exceeding the service target and a time-series breakdown by route. These measures are more actionable than a single overall average.

Outcome and learning points

The median indicates the typical customer experience more effectively than the mean in this skewed sample. The IQR describes the consistency of ordinary deliveries, while the outlier rule identifies an exception requiring investigation. Descriptive statistics supports a decision, but operational knowledge is required to explain the result.

8. Professional practice, technology and limitations

Spreadsheet software and statistical packages can calculate summaries rapidly, but analysts remain responsible for data quality and interpretation. Before calculation, check missing values, duplicated records, impossible values, units, date definitions and the population represented by the data. Automated dashboards should document whether statistics use a sample or full population denominator and which percentile or quartile convention is applied.

AI-assisted analytics can suggest summaries, detect unusual records and generate charts. It should not be allowed to silently remove outliers, infer missing values or present correlation as causation. Human review, reproducible code, versioned data and an audit trail are essential, particularly for performance evaluation, credit decisions or customer segmentation.

Descriptive statistics also raises ethical issues. Aggregated summaries can conceal disparities between regions, demographic groups or customer segments. Analysts should avoid reporting sensitive information when small groups could be re-identified. Definitions, exclusions and transformations should be transparent. A visually attractive chart can still mislead if its axis is truncated, classes are unequal without density adjustment, or percentages use different denominators.

Important limitations include the following:

• A mean can conceal multimodal groups or extreme skewness.
• A median does not show the size of all deviations.
• Standard deviation is not robust to outliers.
• A frequency distribution depends on classification choices.
• A z-score is meaningful only relative to a clearly defined reference distribution.
• Covariance and correlation do not prove causality.
• Small samples can make shape measures unstable.
• Summary statistics can hide changes over time and differences between subgroups.

Best practice is therefore to combine numerical summaries with appropriate visualisations, subgroup analysis, contextual knowledge and explicit documentation.

Chapter Summary

Descriptive statistics organises raw data into interpretable information. Frequency distributions show how observations are distributed across values or classes, while relative and cumulative frequencies support comparison and threshold analysis. The mean uses all observations but is sensitive to extreme values; the median is resistant to extremes; and the mode identifies the most common category or value.

Range, IQR, variance and standard deviation measure dispersion in different ways. The sample variance divides by n − 1, whereas population variance divides by N. Percentiles locate observations, and z-scores express distances from a reference mean in standard deviation units. Skewness describes asymmetry and kurtosis describes tail and concentration characteristics, subject to convention differences. Covariance measures joint movement in original units, while correlation standardises linear association.

Effective business analysis selects measures appropriate to the data, checks assumptions, investigates unusual observations and avoids causal claims from descriptive association alone. Technology can improve speed and consistency, but transparency, privacy, subgroup fairness and professional judgement remain essential.

Key Terms

Observation: One recorded unit in a dataset.

Variable: A characteristic measured or recorded for observations.

Frequency: The number of times a value or category occurs.

Relative frequency: A frequency divided by the total number of observations.

Cumulative frequency: The running total of frequencies up to a specified value or class.

Mean: The arithmetic average of numerical observations.

Median: The middle ordered observation, or the average of the two middle observations.

Mode: The most frequently occurring value or category.

Range: The maximum value minus the minimum value.

Quartile: A value dividing ordered data into position-based sections.

Interquartile range: The difference Q₃ − Q₁, covering the middle 50% of observations.

Variance: The average squared deviation, using a specified population or sample denominator.

Standard deviation: The square root of variance.

Coefficient of variation: Standard deviation expressed relative to the mean.

Skewness: A measure describing distributional asymmetry.

Kurtosis: A measure related to concentration and tail heaviness.

Covariance: A measure of the direction of joint variation between two numerical variables.

Correlation: A unit-free measure of linear association, bounded between −1 and +1.

Z-score: The number of reference standard deviations an observation lies above or below the reference mean.

Outlier: An observation unusually distant from the main body of data under a stated criterion.

Review Questions

Short Answer Questions

1. What is the difference between a frequency and a relative frequency?
2. Why is the median often preferred to the mean for highly skewed income data?
3. Distinguish sample variance from population variance.
4. What does a z-score of −1.5 indicate?
5. Why does covariance depend on measurement units?

Long Answer Questions

1. Explain how frequency distributions, histograms and cumulative frequencies support business decision-making.
2. Compare the mean, median and mode, including an appropriate business use for each.
3. Explain range, IQR, variance, standard deviation and coefficient of variation, including their limitations.
4. Discuss skewness and kurtosis and explain why software definitions should be checked before interpretation.
5. Explain why correlation and covariance cannot by themselves establish causation.

Analytical / Application Questions

1. A retailer records daily sales of 12, 13, 13, 14, 15, 16 and 40 units. Calculate the mean, median, mode and range. Which measure best represents a typical day, and why?
2. Two departments have productivity means of 80 and 120 units. Their standard deviations are 8 and 18 units respectively. Calculate each coefficient of variation and identify which department has greater relative variability.
3. A customer-service analyst finds a strong positive correlation between staff training expenditure and customer satisfaction across branches. Identify at least two reasons why this result does not prove that training caused the satisfaction difference, and propose additional analysis.

Multiple Choice Questions with Answers

Question 1

A dataset contains the values 4, 6, 6, 8 and 11. What is its median?

A. 6

B. 7

C. 8

D. 11

Correct Answer: A. 6

Question 2

Which measure is generally most resistant to an extreme outlier?

A. Mean

B. Median

C. Variance

D. Range

Correct Answer: B. Median

Question 3

If a sample has n = 8 observations, which denominator is used in the usual sample variance?

A. 7

B. 8

C. 9

D. 64

Correct Answer: A. 7

Question 4

A z-score of +2 means that an observation is:

A. Two units above the median

B. Two percentage points above the mean

C. Two standard deviations above the reference mean

D. Twice the reference mean

Correct Answer: C. Two standard deviations above the reference mean

Question 5

The interquartile range is calculated as:

A. Q₁ + Q₃

B. Q₃ − Q₁

C. Maximum − minimum

D. Mean ÷ standard deviation

Correct Answer: B. Q₃ − Q₁

Question 6

A positively skewed distribution commonly has:

A. A longer right tail

B. A longer left tail

C. No variability

D. Equal values only

Correct Answer: A. A longer right tail

Question 7

Which statement about correlation is correct?

A. It proves that one variable causes another

B. It is always greater than 1

C. It measures linear association and is bounded between −1 and +1

D. It is unaffected by measurement units because covariance is not involved

Correct Answer: C. It measures linear association and is bounded between −1 and +1

Question 8

For values 2, 4 and 6, the arithmetic mean is:

A. 3

B. 4

C. 5

D. 6

Correct Answer: B. 4

Question 9

Advertising and sales have a positive sample covariance. This indicates that:

A. They have identical units

B. They necessarily have a causal relationship

C. Above-average values of one tend to occur with above-average values of the other

D. Their correlation must equal 1

Correct Answer: C. Above-average values of one tend to occur with above-average values of the other

Question 10

A coefficient of variation is most appropriately used to:

A. Compare relative variability for positive ratio-scale variables

B. Replace the median for categorical data

C. Measure the direction of covariance

D. Identify the modal category

Correct Answer: A. Compare relative variability for positive ratio-scale variables