Descriptive Statistics and Business Reporting

Introduction

Descriptive statistics provides the language used to convert raw business data into interpretable evidence. A transaction table may contain thousands of sales records, customer ratings, delivery times, or employee observations. Descriptive statistics reduces such data into summaries that reveal the typical value, variation, relative position, distributional shape, and relationships between variables.

The purpose is not merely to calculate numbers. A useful business report must connect a statistic to a decision. For example, an average delivery time may appear acceptable, while the 90th percentile reveals that one customer in ten waits considerably longer. Similarly, a strong correlation between advertising expenditure and sales does not by itself prove that advertising caused the sales increase.

This chapter develops the principal measures of location, dispersion, position, shape, and association. It also explains how to communicate results through tables, dashboards, and decision-focused commentary.

Learning objectives

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

• distinguish populations from samples and parameters from statistics;
• calculate and interpret the mean, median, mode, range, variance, and standard deviation;
• calculate percentiles and quartiles and explain their business meaning;
• interpret skewness and identify the effect of outliers;
• calculate covariance and correlation while recognising their limitations;
• select appropriate summaries for different data types and distributions;
• design a concise statistical report or dashboard for decision-makers;
• identify ethical, data-quality, and governance risks in descriptive reporting.

1. The foundation: data, variables, and measurement

A population is the complete set of units relevant to a study, such as every invoice issued by a company in a year. A sample is a subset selected from that population. A numerical description of a population is a parameter; a numerical description calculated from a sample is a statistic.

Variables may be qualitative or quantitative. Qualitative variables describe categories, such as region or payment method. Quantitative variables express numerical amounts. Discrete variables are countable, such as number of complaints; continuous variables can take values on a continuum, such as delivery time.

Measurement scale affects the statistics that are meaningful:

A data audit should establish the unit of measurement, time period, inclusion rules, missing-value treatment, and whether each row represents a unique business event. A mean calculated from duplicated transactions or mixed currencies can be precise but misleading.

2. Measures of location

Measures of location identify a typical or central value.

2.1 Arithmetic mean

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

Here, x̄ is the sample mean, xᵢ is observation i, and n is the number of observations. The population mean is commonly represented by μ and divides by the population size N. The mean is appropriate for quantitative data when observations are measured on a meaningful numerical scale. It uses every value but is sensitive to extreme observations.

Example. Five daily sales totals, in thousands of dollars, are 8, 10, 11, 12, and 19.

The mean daily sales total is $12,000. This does not mean that any particular day necessarily produced $12,000; it is the balancing point of the observations.

A weighted mean is appropriate when observations contribute unequally:

where wᵢ is the weight assigned to xᵢ. Weights must reflect a defensible basis, such as number of customers represented, and must not be introduced merely to obtain a preferred result.

2.2 Median

The median is the middle ordered value. If n is odd, it is the value at position (n + 1) ÷ 2. If n is even, it is commonly calculated as the average of the values at positions n ÷ 2 and n ÷ 2 + 1. Different software packages may use slightly different percentile conventions, so the selected method should be documented.

For the sales data 8, 10, 11, 12, 19, the median is 11. The median is less affected by unusually high or low values than the mean and is therefore often preferred for house prices, salaries, customer spending, and waiting times.

2.3 Mode

The mode is the most frequently occurring value or category. It is useful for identifying the most common product, payment method, or rating. A dataset may have one mode, several modes, or no repeated value. The mode should not be confused with the mean or median.

2.4 Choosing a measure of centre

Use the mean when the data are quantitative, reasonably symmetric, and not dominated by outliers. Use the median for skewed data or when a typical case is more informative than a balancing point. Use the mode for categorical data or when the most common choice is operationally important. In professional reporting, presenting mean and median together can reveal asymmetry.

3. Measures of dispersion

Location alone is incomplete. Two branches may have the same average sales but very different consistency and risk.

3.1 Range and interquartile range

The range is:

It is simple but depends entirely on two observations. The interquartile range (IQR) is:

where Q₁ is the first quartile, the 25th percentile, and Q₃ is the third quartile, the 75th percentile. The IQR describes the spread of the middle 50% and is relatively resistant to outliers.

3.2 Variance

For a sample, variance is:

For a population, the corresponding formula is:

The sample formula uses n − 1, known as the degrees-of-freedom correction, when sample data are used to estimate population variability. Variance is expressed in squared units, which can make direct interpretation difficult.

Worked example. Consider delivery times of 2, 4, 4, 6, and 9 minutes. The mean is 5 minutes. Deviations from the mean are −3, −1, −1, 1, and 4; their squared deviations are 9, 1, 1, 1, and 16. The sum is 28.

If these five observations represent the entire population, population variance is 28 ÷ 5 = 5.6 minutes². The choice depends on whether the records are the complete target population or a sample intended to represent a larger population.

3.3 Standard deviation

Standard deviation is the square root of variance:

For the sample above, s = √7 ≈ 2.65 minutes. Unlike variance, standard deviation has the same units as the original variable. It represents a typical distance from the mean, although the exact interpretation depends on the distribution’s shape.

For approximately bell-shaped data, the empirical rule suggests that about 68% of observations lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three. These percentages are not universal laws and should not be applied automatically to highly skewed data.

3.4 Coefficient of variation

The coefficient of variation compares relative variability:

It is meaningful when the mean is positive and the measurement scale has a meaningful zero. For example, it may compare demand volatility across products with different average sales. It should not generally be used for variables such as temperature in Celsius, where zero is arbitrary.

4. Measures of position

Position measures show where an observation lies relative to the rest of the dataset.

A percentile is a value below which a specified percentage of observations fall. The 90th percentile of delivery time is the value at or below which approximately 90% of deliveries are completed. Percentiles are especially useful for service-level commitments and risk thresholds.

Quartiles divide ordered data into four sections: Q₁, the 25th percentile; Q₂, the median; and Q₃, the 75th percentile. Deciles divide data into ten sections.

Percentile calculations vary by convention. A common interpolation method assigns the rank:

where p is the desired percentile expressed as a decimal. If L is not an integer, the percentile is interpolated between adjacent ordered observations. Because other methods use different rank formulas, reports should state the software or convention used.

The standard score, or z-score, expresses an observation relative to the mean:

For a population, μ and σ replace x̄ and s. A positive z-score indicates a value above the mean; a negative score indicates a value below it. Z-scores have no units and are useful for comparing variables measured on different scales, provided the standard deviations are meaningful.

The 1.5 × IQR rule is a screening convention for possible outliers. Values below Q₁ − 1.5IQR or above Q₃ + 1.5IQR are flagged. A flagged value is not automatically an error; it may represent a legitimate high-value customer or an unusual but important event.

5. Distribution shape

5.1 Symmetry and skewness

A symmetric distribution has similar shape on both sides of its centre. In a positively skewed distribution, a long tail extends toward larger values; commonly, mean > median. Income, claim amounts, and customer spending are often positively skewed. In a negatively skewed distribution, the tail extends toward smaller values; commonly, mean < median.

A frequently used sample moment coefficient of skewness is:

where n is sample size, x̄ is the sample mean, and s is sample standard deviation. Software may use alternative bias corrections or definitions. Therefore, the numerical value should be accompanied by a histogram, box plot, or percentile summary rather than interpreted in isolation.

Skewness has practical consequences. The mean may overstate a typical customer’s spending in a right-skewed dataset. A median, IQR, and upper percentile may provide a better management summary. Transformations such as logarithms can make highly skewed data easier to model, but transformed results must be translated back into business units before reporting.

5.2 Kurtosis and tails

Kurtosis concerns tail heaviness and concentration relative to a reference distribution. Heavy tails imply more extreme observations than expected under a normal model. In operational reporting, the most important question is often not the exact kurtosis coefficient but whether rare events create financial, safety, or service risk.

6. Association between two variables

6.1 Covariance

Sample covariance measures whether two variables tend to move together:

Here, x̄ and ȳ are sample means. Positive covariance indicates that above-average values of X tend to occur with above-average values of Y; negative covariance indicates the opposite. Covariance depends on the units of both variables, so its magnitude is difficult to compare across datasets.

6.2 Pearson correlation

Pearson’s sample correlation standardises covariance:

where sₓ and sᵧ are the sample standard deviations. Equivalently:

The value lies between −1 and +1. Values near +1 indicate strong positive linear association, values near −1 strong negative linear association, and values near 0 little linear association. Correlation is unit-free.

Correlation does not establish causation. A relationship may result from reverse causality, a third variable, selection bias, common time trends, or coincidence. Pearson correlation is also sensitive to outliers and may be inappropriate for strongly nonlinear relationships. A scatter plot should accompany a reported correlation whenever possible.

7. Decision-focused statistical reporting

A report should answer three questions: What happened? How certain or variable was it? What decision follows?

A useful summary table may include count, missing values, mean, median, standard deviation, quartiles, and selected percentiles. The required statistics depend on the decision. A call-centre manager may need median wait time, 90th percentile wait time, and the percentage breaching a service target rather than only the mean.

A concise dashboard should contain:

1. A clear purpose, such as monitoring delivery performance.
2. A defined population and time period.
3. A small number of relevant indicators.
4. Comparisons, such as target versus actual or current period versus prior period.
5. Distributional evidence, such as a box plot or percentile chart.
6. Filters with visible selections.
7. Annotations explaining unusual events.
8. Data freshness, definitions, and source information.

Good visual choices include a histogram for distribution, a box plot for group comparison and outliers, a line chart for time trends, and a scatter plot for association. Pie charts are generally less effective for comparing many categories. Colour should communicate meaning, not decoration, and dashboards should remain interpretable for colour-blind users.

A report should avoid false precision. If data are recorded to the nearest minute, reporting 7.384 minutes implies unsupported accuracy. Percentages should include denominators, and small groups should be labelled because their statistics may be unstable.

8. Illustrative case: improving delivery performance

Background

A fictional online retailer promises delivery within two days. The operations manager wants to know whether the promise is being met consistently across 12 recent deliveries. Delivery times, in days, are:

1.2, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.3, 2.6, 4.8.

Problem

The mean is approximately 2.07 days, while the median is 1.85 days. The mean is pulled upward by the 4.8-day delivery. The range is 3.6 days, indicating substantial spread. The retailer needs to know whether the average conceals a service problem.

Analysis

The 4.8-day observation is well above the upper portion of the ordered data and should be investigated. It may be a genuine remote-area delay, a recording error, or an exceptional logistics failure. It should not be deleted merely because it is inconvenient.

The median indicates that a typical delivery is within two days, but the upper tail indicates customer risk. A manager should calculate the percentage meeting the promise, the 90th percentile, and results by warehouse, carrier, and region. A box plot can reveal whether the long delay is isolated or part of a recurring pattern.

Approach and solution

The operations team validates the record, confirms that it is genuine, and adds a dashboard containing median delivery time, percentage delivered within two days, 90th percentile delivery time, and a trend by carrier. A separate alert identifies deliveries exceeding three days. The team then examines route capacity and handoff delays rather than changing the definition of an on-time delivery.

Outcome

The analysis shows that the overall mean alone would have encouraged complacency. The median describes typical performance, while the upper percentile and exception count expose the customer experience at the tail. Management can therefore target the specific carrier and route responsible for repeated delays.

Learning points

• Mean and median answer different questions.
• Outliers require investigation, not automatic removal.
• Service decisions often depend on percentiles and thresholds.
• A dashboard should connect measures to operational action.
• Definitions and denominators must be visible to users.

9. Limitations, ethics, and governance

Descriptive statistics summarises observed data; it does not automatically explain causes or predict future outcomes. A biased sample can produce biased summaries. Missing values may be systematic—for example, dissatisfied customers may be less likely to complete a survey. Aggregating data can conceal differences between regions, demographic groups, or customer segments.

Analysts should protect personal information through data minimisation, access controls, aggregation, and appropriate retention. Sensitive categories should not be exposed unnecessarily in dashboards. Small-cell suppression may be required where reporting could identify individuals. Definitions should be version-controlled so that a changed metric is not mistaken for a real performance change.

AI-enabled analytics can automate anomaly detection, narrative generation, and dashboard recommendations. However, generated explanations must be checked against source data. Automation may amplify biased data, produce unsupported causal claims, or obscure how a statistic was calculated. Human review, audit trails, reproducible code, and clear ownership remain essential.

Best-practice checklist

Before publishing a statistical report, confirm that:

• the population, sample, time period, and units are stated;
• missing and duplicate records have been assessed;
• the measure of centre matches the distribution and decision;
• dispersion and relevant percentiles are included;
• outliers have been investigated;
• charts show appropriate scales and labels;
• correlation has not been presented as causation;
• denominators and small sample sizes are visible;
• privacy and access requirements are satisfied;
• calculations can be reproduced and definitions are documented.

Chapter Summary

Descriptive statistics transforms raw observations into evidence for business decisions. Measures of location—mean, median, and mode—describe typical or common values, while range, IQR, variance, standard deviation, and coefficient of variation describe dispersion. Percentiles, quartiles, and z-scores describe relative position. Skewness and tail behaviour influence which summaries best represent a dataset. Covariance indicates joint movement, and Pearson correlation measures the strength and direction of linear association, but neither proves causation.

Professional reporting requires more than accurate arithmetic. Analysts must define the data, select statistics appropriate to the measurement scale and distribution, communicate uncertainty and variation, use effective visualisations, investigate outliers, and protect privacy. A decision-focused dashboard connects a small set of transparent indicators to an explicit managerial action.

Key Terms

Population: The complete set of units relevant to a study.

Sample: A subset of a population.

Parameter: A numerical characteristic of a population.

Statistic: A numerical characteristic calculated from a sample.

Mean: The arithmetic average of quantitative observations.

Median: The middle ordered value.

Mode: The most frequently occurring value or category.

Range: The maximum value minus the minimum value.

Variance: The average squared deviation, using the appropriate population or sample denominator.

Standard deviation: The square root of variance, expressed in the original units.

Quartile: A value dividing ordered data into four sections.

Percentile: A value below which a specified percentage of observations fall.

Interquartile range: The difference between the third and first quartiles.

Z-score: A standardised distance from the mean.

Skewness: A measure of asymmetry in a distribution.

Covariance: A measure of joint variation between two variables.

Correlation: A standardised measure of linear association.

Outlier: An observation unusually distant from the main body of data.

Dashboard: A visual display of selected metrics for monitoring or decision-making.

Review Questions

Short Answer Questions

1. Distinguish a population parameter from a sample statistic.
2. Why is the median often preferred to the mean for highly skewed income data?
3. What does the interquartile range measure?
4. Explain the difference between covariance and correlation.
5. Why does correlation not establish causation?

Long Answer Questions

1. Explain how mean, median, mode, range, variance, and standard deviation provide complementary information about a business dataset.
2. Describe the assumptions and interpretation of the sample variance formula.
3. Explain how skewness and outliers should influence the design of a management report.
4. Discuss the features of an effective decision-focused statistical dashboard.
5. Evaluate the ethical and governance risks involved in publishing customer-level descriptive statistics.

Analytical / Application Questions

1. A store records weekly revenues of 20, 21, 22, 23, and 44 thousand dollars. Calculate the mean and median, identify the likely effect of the extreme value, and recommend which measure should headline a report.
2. A call centre reports an average waiting time of four minutes and a 90th percentile of eleven minutes. Explain why both figures matter and recommend two operational actions.
3. Advertising expenditure and sales have a strong positive correlation over six months. Identify at least three reasons why management should not conclude that advertising alone caused the sales increase, and propose additional analysis.

Multiple Choice Questions with Answers

Question 1

Which measure is generally most resistant to an extreme outlier?

A. Mean

B. Median

C. Variance

D. Range

Correct Answer: B. Median

Question 2

A sample contains 2, 4, 4, 6, and 9. What is its sample variance?

A. 5.6

B. 7

C. 2.65

D. 28

Correct Answer: B. 7

Question 3

What does the interquartile range represent?

A. The spread of all observations

B. The difference between the mean and median

C. The spread of the middle 50% of observations

D. The percentage above the mean

Correct Answer: C. The spread of the middle 50% of observations

Question 4

A positively skewed distribution commonly has which relationship?

A. Mean less than median

B. Mean equal to zero

C. Mean greater than median

D. Mean equal to the mode in every case

Correct Answer: C. Mean greater than median

Question 5

Which statistic is unit-free and bounded between −1 and +1?

A. Covariance

B. Variance

C. Standard deviation

D. Pearson correlation

Correct Answer: D. Pearson correlation

Question 6

If a delivery time has a z-score of −2, what does this indicate?

A. It is two standard deviations below the mean

B. It is two units below the median

C. It is twice the mean

D. It is two percent below the mean

Correct Answer: A. It is two standard deviations below the mean

Question 7

Which chart is most directly suitable for examining the relationship between advertising expenditure and sales?

A. Scatter plot

B. Pie chart

C. Histogram of product categories

D. Single-value gauge

Correct Answer: A. Scatter plot

Question 8

Why is the sample variance denominator commonly n − 1 rather than n?

A. To convert units into percentages

B. To apply a degrees-of-freedom correction when estimating population variability

C. To remove all outliers

D. To make variance equal to standard deviation

Correct Answer: B. To apply a degrees-of-freedom correction when estimating population variability

Question 9

A 90th percentile delivery time of 11 minutes means that approximately:

A. 90% of deliveries take exactly 11 minutes

B. 10% of deliveries take 11 minutes

C. 90% of deliveries take no longer than about 11 minutes

D. The mean delivery time is 11 minutes

Correct Answer: C. 90% of deliveries take no longer than about 11 minutes

Question 10

Which statement is the best reporting practice?

A. Report only the mean to keep the dashboard simple

B. Remove every outlier before calculating statistics

C. Show definitions, denominators, units, and relevant measures of variation

D. Treat a strong correlation as proof of causation

Correct Answer: C. Show definitions, denominators, units, and relevant measures of variation