Descriptive Statistics for Finance Professionals: Mean, Variance and Standard Deviation

Descriptive statistics summarise a set of numbers: where it is centred, how spread out it is, and what shape it has. For finance professionals, the same ideas sit behind everyday questions. What is a typical monthly cost? How variable is it? Is a single large item distorting the average? This guide explains the mean, median, variance and standard deviation in plain terms, using definitions from the NIST/SEMATECH e-Handbook of Statistical Methods. NIST is a US body, so this is a US reference work on method, not Irish guidance, and the handbook pages we opened show no date. All worked numbers below are invented for illustration.

The NIST news release (15 June 2002, page updated 3 February 2025) says the e-Handbook is aimed at engineers, scientists, businesses, researchers and teachers who use statistical techniques in their work. If you later want to apply these ideas with code, our guide to Python for accountants shows where data analysis tooling can take you.

Location: mean and median

The handbook's Measures of Location page defines the mean as the sum of the data points divided by the number of data points, and says it is the value most commonly referred to as the average. It notes the mean is sensitive to outliers: extreme values in the tails distort it.

The median is the middle value, with half the data smaller and half larger; for an even count it is the average of the two middle values. The page says that for data with extreme values in the tails, the median provides a better estimate of location than the mean, because it is based on ranks. It also notes that for skewed data it is not obvious whether the mean or the median is more meaningful.

Illustrative example (invented numbers, in thousands of euro): five monthly costs of 10, 12, 11, 13 and 14 have a mean of 12 and a median of 12. Now suppose a sixth month contains a one-off item and comes in at 60. The mean becomes 20 (120 divided by 6), while the median is 12.5 (the average of 12 and 13). One unusual month moves the mean a long way and the median barely at all. Our suggestion: when a report quotes an average, check whether a single large item is driving it.

The page also describes the mid-range, which is (smallest + largest) divided by two, and says its use is typically restricted to situations where behaviour at the extreme points is relevant.

Scale: variance and standard deviation

Location tells you the centre; scale tells you the spread. The handbook's Measures of Scale page defines the variance as:

s² = Σ(Y_i − Ȳ)² / (N − 1), where the bar over Y is the mean of the data.

The page describes variance as roughly the arithmetic average of the squared distance from the mean. It defines the standard deviation as the square root of the variance, and says the standard deviation restores the units of the spread to the original data units.

The page uses N − 1 in the denominator. In the text we read, it does not say whether the data are a sample or a population, or why N − 1 is used. Because software can offer both conventions, check which one a tool uses before comparing results from different tools.

Illustrative example (invented numbers, same five months): the mean is 12. The differences from the mean are −2, 0, −1, 1 and 2. Squared, they are 4, 0, 1, 1 and 4, which add to 10. Dividing by N − 1, that is 4, gives a variance of 2.5. The square root is about 1.58, so the standard deviation is roughly 1.6 (thousand euro). Reading it back: a typical month in this invented set sits within about 1.6 thousand euro of the average.

Why use the standard deviation rather than the variance? Because the variance is in squared units (squared euro), which is hard to interpret. Taking the square root returns to euro, as the NIST page says.

Shape: histograms and skewness

Numbers such as the mean and standard deviation compress a dataset. A picture can show what they hide. The handbook's Histogram page says a histogram graphically shows the centre (location), spread (scale), skewness and presence of outliers in the data.

The handbook's page on skewness and kurtosis defines skewness using cubed deviations from the mean divided by s cubed, and notes that many software programs compute an adjusted version that corrects for sample size. For kurtosis, it says the standard normal distribution has a kurtosis of 3 under its definition, that some sources subtract 3 to give excess kurtosis, and that which definition is used is a matter of convention. The page warns that software output may follow either convention, so users should check which applies.

A practical approach: whenever you quote a skewness or kurtosis figure from software, note which definition the software uses, and do not compare figures across tools without checking.

Using these measures in finance work

These are our suggestions on application, not statements from the NIST pages:

• Report spread alongside the average. Two cost lines can share an average yet differ greatly in variability, and a budget holder will want to know which one is steadier.
• Look at the data before summarising it. A histogram can reveal a skewed pattern or an outlier that a single average hides.
• Investigate outliers; do not just delete them. An outlier may be an error, or it may be a real one-off event that management needs to hear about.
• Keep small samples in perspective. A standard deviation from a handful of months is a rough guide only.

Descriptive statistics also underpin forecasting and modelling. Our guide to financial modelling in Excel for Irish finance teams covers how assumptions about variability enter a model, and our overview of machine learning for accountants shows where these basics lead next.

A limit worth stating

The NIST pages define these measures; they do not tell you which to report in a given business setting, and they are not finance guidance. Where a regulator, a professional body or your own policy specifies how a figure must be calculated, follow that.

Learnsignal's offices are off Baggot Street in Dublin, and we offer in-person training from there. For current details, see our CPD pages.

Frequently asked questions

What is the difference between variance and standard deviation?

According to the NIST handbook, the standard deviation is the square root of the variance, and taking the square root restores the units of the spread to the original data units.

When is the median better than the mean?

The NIST Measures of Location page says that for data with extreme values in the tails, the median provides a better estimate of location than the mean.

Why does the variance formula divide by N minus 1?

The page we read uses N − 1 but does not explain why or say whether the data are a sample or population. Check the documentation of any tool you use to see which convention it applies.

Are the example numbers real?

No. Every figure in the worked examples is invented and rounded for illustration, and none represents real market or company data.

Mean and median describe the centre, variance and standard deviation describe the spread, and a histogram shows the shape; reporting all three gives a far more honest summary than an average on its own.