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Fix the driver behind crashes, sound loss and screen glitchesFind Drivers →Clear out junk files and repair common Windows errorsFree Scan →To summarize a numeric dataset, report its observation count and units, describe its center with the mean, median, and—when useful—mode, then describe its spread with a clearly labeled standard deviation. These statistics answer different questions: the mean and median locate the center in different ways, the mode identifies the most frequent value, and standard deviation measures variation around the mean.
Start with the variable, count, and units
Before calculating, identify what each observation measures, how many observations there are, and the units. For example, a summary of delivery times should say whether values are in minutes and how many deliveries were included. Those details give the statistics context and make them interpretable.
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For a small dataset, write the values in order first. For larger datasets, a calculator or statistics program can calculate descriptive statistics; an OpenStax example shows descriptive output that includes count, mean, standard deviation, minimum, quartiles, and maximum: OpenStax, “Data Science with Python”.
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Mean
Add all observations and divide by their count. The mean uses every value, so unusually high or low observations can pull it toward an extreme. It is useful when the arithmetic balance of all observations matters and those extremes should influence the summary. OpenStax explains this sensitivity in its discussion of measures of center: OpenStax, “Measures of Center”.
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Median
Sort the observations. With an odd count, the median is the middle value; with an even count, it is the average of the two middle values. The median is less affected by extreme values than the mean, so it can better represent the center of skewed data or data with unusually large or small observations.
Mode
The mode is the value that occurs most often. A dataset may have more than one mode, or no repeated value that makes a mode useful. It is especially helpful when the most common category or value is itself the question; unlike the mean, the mode can describe categorical data.
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- This guide is a perfect overview for the topics covered in introductory statistics courses.
Calculate standard deviation—and label the convention
Standard deviation describes the amount of variation around the mean, in the same units as the observations. A smaller standard deviation means values are more concentrated near the mean; a larger one means they are more spread out. It is not the same as variance, which uses squared units. OpenStax describes standard deviation as a numerical measure of overall variation in the dataset, in the data’s units: OpenStax, “Measures of Variation”.
Use the sample formula when the observations are a sample and the calculation is intended to estimate variability in a broader population. Use the population formula when the data include the complete population of interest. The denominators differ, so state which convention you used:
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- Sample standard deviation: s = √[Σ(xi − x̄)² / (n − 1)], where x̄ is the sample mean and n is the sample size. OpenStax provides this sample formula in its formula review.
- Population standard deviation: σ = √[Σ(xi − μ)² / N], where μ is the population mean and N is the population size. OpenStax distinguishes the sample and population measures and their uses in its discussion of spread.
Standard deviation summarizes spread relative to the mean. It should not be interpreted on its own when the distribution is strongly skewed or includes unusual observations.
Worked example: 2, 4, 4, 5, 10
The dataset has five observations and is already ordered. Its sum is 25, so the mean is 25 ÷ 5 = 5. The middle value is 4, making the median 4; 4 appears most often, so the mode is also 4.
The squared deviations from the mean are 9, 1, 1, 0, and 25; their sum is 36. For these five values treated as the complete population, the standard deviation is √(36 ÷ 5) ≈ 2.68. Treated as a sample, it is √(36 ÷ 4) = 3. The value 10 pulls the mean above the median, illustrating why both measures may help explain a dataset.
Choose a summary that fits the data
| Measure | What it describes | When it is useful |
|---|---|---|
| Mean | Arithmetic center using every observation | When all values, including extremes, should affect the center |
| Median | Middle of the ordered observations | When skew or extreme values could make the mean unrepresentative |
| Mode | Most frequent value or category | When the most common value or category matters |
| Standard deviation | Spread around the mean, in the original units | When describing variation, with sample or population convention stated |
Do not treat these as interchangeable versions of “the average.” Report multiple measures when they add complementary information. A difference between mean and median can signal that the measures respond differently to the distribution, but a summary alone cannot reveal every cluster, gap, skew, or unusual value.
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Report enough context to make the numbers useful
A compact descriptive summary can include the variable and its units, count, mean, median, mode when informative, standard deviation with its sample or population label, minimum, quartiles, and maximum. Add a short explanation of whether the mean and median differ meaningfully and whether the distribution or unusual observations affect interpretation. When comparing groups, use the same statistics and standard-deviation convention for each; center and spread may still not capture differences in distribution shape.
A graphing calculator’s one-variable statistics function can provide summary values; OpenStax describes that workflow in its discussion of spread: OpenStax, “Measures of the Spread of the Data”.
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