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Stat Dojo / Foundations

Descriptive Statistics

Mean, median, mode, range, variance, and standard deviation, and when each one misleads, with Excel and Minitab.

Question it answers
What is a typical value, and how much do the values vary?
Data needed
A column of numerical measurements
Key output
Center, spread, and shape, with a plot
Default pair
Mean and SD (symmetric); median and IQR (skewed)
Assumptions
None for describing; shape decides which summary fits
Excel
AVERAGE, MEDIAN, STDEV.S, QUARTILE.EXC; Descriptive Statistics tool
Minitab
Stat > Basic Statistics > Display Descriptive Statistics
Why it matters
Every test and chart builds on these numbers

The Idea in Plain Language

Descriptive statistics boil a pile of numbers down to a few that describe it. Two jobs matter most: where the data are centered (location) and how spread out they are (variation). A third, the shape, tells you whether the first two can be trusted.

JobMeasuresTells you
CenterMean, median, modeA typical value
SpreadRange, interquartile range, variance, standard deviationHow much the values differ
ShapeSkewness, kurtosis, and a pictureSymmetric or lopsided, heavy tails or not
PositionMinimum, quartiles, percentiles, maximumWhere a value sits in the data
Why it matters. Every later method, from a control chart to a t-test, is built from these numbers. A wrong or misleading summary at the start feeds wrong conclusions all the way down.

How Each Measure Works

MeasureFormula or ruleUse it when
Meanx̄ = Σx / nData are roughly symmetric; it uses every value
MedianMiddle value of the sorted dataData are skewed or have outliers
ModeMost frequent valueCategories or discrete values
RangeMaximum − minimumVery small samples; subgroup charts
Interquartile rangeQ3 − Q1Spread that ignores the extremes
Sample variances² = Σ(x − x̄)² / (n − 1)Intermediate step; additive across independent sources
Sample standard deviations = √s²The standard measure of spread, in the data’s own units
Standard error of the means / √nHow precisely the mean is known
Coefficient of variations / x̄ × 100%Compare spread between things with different averages
Why divide by n − 1? The deviations are measured from the sample mean, which sits closer to the data than the true mean would. That makes them a little too small, and dividing by n − 1 instead of n corrects the bias. The population formula, which divides by n, is for when you have every item, not a sample.

A useful rule for roughly bell-shaped data: about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three.

Worked Example 1: By Hand

Eight order cycle times in minutes: 12, 15, 14, 10, 13, 16, 11, 13.

xx − mean(x − mean)²
12-1.001.0000
15+2.004.0000
14+1.001.0000
10-3.009.0000
13+0.000.0000
16+3.009.0000
11-2.004.0000
13+0.000.0000
Σx = 1040.0028.0000
  1. Mean = 104 / 8 = 13.000.
  2. Median: sorted values are 10, 11, 12, 13, 13, 14, 15, 16; the middle two are 13 and 13, so the median is 13.
  3. Range = 16 − 10 = 6.
  4. Variance = 28.0000 / (8 − 1) = 4.0000.
  5. Standard deviation = √4.0000 = 2.000 minutes.
Conclusion. Orders take 13.00 minutes on average with a standard deviation of 2.00 minutes. The mean and median agree (13.00 and 13), so the data are close to symmetric and the mean is a fair summary. Check: the deviations always sum to zero.

Worked Example 2: When the Mean Misleads

A larger sample of 24 order processing times, in minutes. Most orders take 12 to 19 minutes, but a few take much longer.

10 14 18 22 26 30 34 38 42 Median 15.1 Mean 17.2 Processing time (minutes)
The slow orders on the right pull the mean above the median. The median stays with the bulk of the data.
StatisticAll 24 ordersWithout the slowest
Mean17.1516.11
Median15.0514.90
Standard deviation6.333.86
Range29.216.1
10 14 18 22 26 30 34 38 42 All 24 orders Without the slowest Minutes box = middle 50%, line = median, diamond = mean, circle = outlier
The box plot shows the same story: one value is far beyond the whiskers, and removing it changes the mean far more than the median.
  • Q1 = 13.65, Q3 = 18.17, interquartile range = 4.52.
  • Trimmed mean (10% off each end) = 15.93, between the median and the mean.
  • Skewness = 2.71: a long right tail. Excess kurtosis = 8.60: heavy tails.
  • Standard error of the mean = 1.29; coefficient of variation = 37%.
Conclusion. For these data the median (15.1 minutes) describes a typical order better than the mean (17.2). Report both, along with the interquartile range (4.5) and the fact that a few orders take far longer. The slow orders are the real finding, and the next step is to ask why they are slow, not to discard them.

Run It in Excel and Minitab

ExcelStep by step

  1. Put the data in a column. Use =AVERAGE(A2:A25), =MEDIAN(A2:A25), and =MODE.SNGL(A2:A25).
  2. Spread: =STDEV.S(A2:A25) (6.327), =VAR.S(A2:A25) (40.03), =MAX()-MIN(). Use STDEV.P only for a full population.
  3. Quartiles: =QUARTILE.EXC(A2:A25, 1) (13.65) matches Minitab. QUARTILE.INC uses a slightly different rule and can differ.
  4. Shape: =SKEW(A2:A25) (2.71), =KURT(A2:A25) (8.60), and =TRIMMEAN(A2:A25, 0.2).
  5. All at once: Data > Data Analysis > Descriptive Statistics, tick Summary statistics.

MinitabStep by step

  1. Stat > Basic Statistics > Display Descriptive Statistics. Choose the column as Variable.
  2. Click Statistics to choose what to show: mean, SE of mean, standard deviation, variance, coefficient of variation, minimum, Q1, median, Q3, maximum, range, IQR, skewness, kurtosis, and more.
  3. Click Graphs to add a histogram, an individual value plot, or a box plot.
  4. For several groups, enter a By variable.
  5. Quicker: Assistant > Graphical Analysis > Graphical Summary or Stat > Basic Statistics > Graphical Summary gives the numbers, a histogram, a box plot, and intervals together.
Minitab session window: Display Descriptive Statistics (typed excerpt, simplified)
Descriptive Statistics: Time

Variable   N  N*   Mean  SE Mean  StDev  Minimum     Q1  Median     Q3  Maximum
Time      24   0  17.15     1.29   6.33    11.80  13.65   15.05  18.17    41.00

Variable  Variance  CoefVar  Range    IQR  Skewness  Kurtosis
Time        40.03    36.89  29.20  4.52      2.71      8.60

Reading and Reporting

  1. Plot first. Look at the shape before trusting any single number.
  2. Pair a center with a spread: mean with standard deviation for symmetric data, median with interquartile range for skewed data.
  3. Give the sample size, and the units.
  4. Show outliers, do not hide them. Investigate the cause before deciding to set one aside, and say what you did.
A sentence you can use. Median order processing time was 15.1 minutes (IQR 13.7 to 18.2, n = 24); the mean of 17.2 minutes is higher because two orders took more than 25 minutes.

Common Mistakes

MistakeWhy it misleadsBetter
Reporting only the meanHides spread and shapeReport the mean and the standard deviation, and plot the data
Using the mean for skewed dataA few large values drag it away from the typical valueUse the median and IQR
Using STDEV.P on a sampleUnderestimates the spreadUse STDEV.S for samples
Deleting an outlier because it is inconvenientIt may be the most important pointFind the cause; document the decision
Comparing standard deviations of different-scale thingsA 2-gram spread and a 2-kilogram spread are not comparableUse the coefficient of variation
Mixing subgroups in one summaryTwo processes blend into a misleading averageSummarize each group, then compare

Try It Yourself

Five fill weights in grams: 498, 502, 500, 503, 497.

  • Find the mean, the median, and the sample standard deviation.
  • What would the standard deviation be if you wrongly divided by n?
Show the answer

Mean = 500 g. Median = 500 g. Sample standard deviation = 2.550 g.

Dividing by n gives 2.280 g, which is smaller. That is the population formula, and it understates the spread of a sample.

Descriptive Statistics: Frequently Asked Questions

Should I use the mean or the median?

Use the mean for symmetric data and the median for skewed data or data with outliers. If they differ a lot, the data are skewed, and that is itself worth reporting.

What is the difference between standard deviation and variance?

Variance is the average squared deviation, and the standard deviation is its square root. The standard deviation is in the same units as the data, so it is easier to interpret. Variance is easier to work with when adding independent sources of variation.

What is the difference between STDEV.S and STDEV.P?

STDEV.S divides by n − 1 and estimates the spread of a population from a sample. STDEV.P divides by n and is for when you have the entire population. Use STDEV.S almost always.

Why does Excel give a different quartile from Minitab?

There are several ways to compute quartiles. QUARTILE.EXC matches Minitab; QUARTILE.INC, and the older QUARTILE function, use a slightly different rule. The difference is small for large samples.

Is a coefficient of variation always useful?

It is useful for ratio-scale data that cannot be negative, such as times and weights, when you want to compare spread across different averages. It makes no sense for data that can be zero or negative, such as temperature in degrees Celsius.

Sources and Further Reading

  • NIST/SEMATECH, e-Handbook of Statistical Methods, Exploratory Data Analysis (itl.nist.gov/div898/handbook).
  • David S. Moore, George P. McCabe, and Bruce A. Craig, Introduction to the Practice of Statistics, Freeman.
  • Rick J. Hyndman and Yanan Fan, “Sample Quantiles in Statistical Packages,” The American Statistician, 1996.
  • Minitab Support, “Methods and formulas for Display Descriptive Statistics” (support.minitab.com).

This content is educational. Worked examples use made-up data. Menu names for Minitab follow recent versions of Minitab Statistical Software and can differ slightly in older releases; Excel steps use Microsoft 365 and the Analysis ToolPak.