Module 02 · Describing data
Combine a picture with summaries of center and spread. See what changes when one observation becomes unusually large.
A list of numbers hides shape. A histogram groups quantitative observations into adjacent numerical intervals and displays their frequencies. Look for concentration, gaps, multiple peaks, skewness, and unusual values. The horizontal scale has numerical meaning; a bar chart for categories instead uses separate category labels.
A time plot is often more informative when order matters. A weekly demand series can have a stable overall histogram while still showing a trend or seasonality. A scatterplot helps investigate two quantitative variables. Choose the display from the question rather than from the software’s default.
The editable data below are fictional handling times in minutes. Change the number of histogram bins while holding the data fixed. The display changes because the grouping changes; the mean, median, and standard deviation do not. A visual impression based on one arbitrary grouping needs care.
Edit the observations or use a preset. Then compare the mean, median, spread, and histogram.
Illustrative handling times in minutes. Histogram bins include their left endpoint; the final bin also includes the maximum. Vertical lines mark the mean and median.
Changing the number of bins changes the display, but not these numerical summaries. Compare the mean and median before and after the long-time preset; inspect unusual values before deciding how to handle them.
10 observations: mean 19.00, median 19.00, sample standard deviation 4.94, first quartile 15.25, third quartile 21.75, and interquartile range 6.50 minutes. Minimum 12.00, maximum 28.00.
The arithmetic mean, x̄ = Σxᵢ / n, uses the magnitude of every value. The median uses the order: sort the data, then select the middle value or average the two middle values. These summarize different aspects of a dataset.
For the illustrative values 2, 4, 4, 4, 5, 5, 7, 9, the sum is 40 and n = 8, so the mean is 5. The middle pair is 4 and 5, giving median 4.5. Replacing 9 with 29 raises the sum by 20 and the mean by 20 / 8 = 2.5, while the median remains 4.5.
For total workload or cost, the mean may be directly relevant even when the data are skewed. For the experience of a typical customer, the median may be more informative. Neither is automatically the correct answer to every question. State which summary you use and why.
The sample variance averages squared deviations from the sample mean using denominator n − 1. The sample standard deviation s is its square root and returns the result to the original units. For the eight values above, the squared deviations sum to 32, so s² = 32 / 7 ≈ 4.571 and s ≈ 2.138.
The interquartile range is Q₃ − Q₁ and summarizes the width of the middle half of the data. Unlike standard deviation, it uses only selected ranks and is less sensitive to one distant observation. The range, maximum minus minimum, depends entirely on the two endpoints.
Quartile conventions differ among software packages, especially for small samples. This explorer uses linear interpolation at index (n − 1)p, counting the first sorted value as index zero. For the eight-value example, Q₁ = 4 and Q₃ = 5.5, so IQR = 1.5. A different documented convention can legitimately give different quartiles.
Standard deviation describes variability among observations. It is not the standard error of a sample mean, which concerns variability across repeated samples. A wide distribution of delivery times can coexist with a precisely estimated mean when the sample is large and well designed.
An outlier is a signal to investigate, not permission to delete a point. Check for entry errors, changes in units, a different operating regime, or a real exceptional event. Report the reason for any exclusion and examine whether the substantive conclusion changes.
For roughly symmetric data without extreme values, mean and standard deviation often give a useful pair. For strongly skewed data, also report the median and IQR. In either case, show the distribution, retain units, and avoid describing a sample summary as proof about an entire population. Descriptive statistics summarize the observed data; inference needs additional design and modeling assumptions.
Try each question before revealing the solution.
The original mean and median are both 10. The replacement adds 20 to the total, so the new mean is 10 + 20 / 5 = 14. The middle sorted observation remains 10, so the median does not change. The gap illustrates the mean’s sensitivity to the magnitude of an extreme observation.
Multiplying every value by 60 multiplies the mean and standard deviation by 60. It multiplies the variance by 60² = 3,600 because variance uses squared deviations. The standardized shape is unchanged. This is why standard deviation, unlike variance, is expressed in the same units as the observations.
The observed centers agree, but the second department’s observations are more dispersed around that center. The means alone hide this difference. You cannot infer the exact proportion above a deadline without seeing the distributions or specifying a suitable model, and the summaries alone do not establish the cause of the difference.
Further reading