Summary Statistics: Calculate and Choose

A useful summary is both a calculation and a choice. A mean answers a different question from a median, and a range responds differently to an extreme observation than an interquartile range. Keep the distribution and context in view while calculating.

Center and Position

Two common measures of center are the mean and median. For a sample of $n$ values, the mean adds every value and divides by their number:

Here $x_i$ is the ith sample data value. Calculate by hand from a sum, or enter a complete data list into technology and request its mean. Check that the list’s reported count matches the number of observations.

Order the data from smallest to largest to find the median. For an odd number of values, the middle observation is a common median. For an even number, average the two middle observations. The ordered smallest and largest observations are the minimum and maximum.

For the ordered eight values 4, 6, 8, 10, 12, 14, 16, 18 minutes, the sum is 88 and the mean is 11 minutes. The median is (10+12)/2=11 minutes; minimum is 4 and maximum is 18. The lower-half median is $Q_1=(6+8)/2=7$ minutes and the upper-half median is $Q_3=(14+16)/2=15$ minutes. The second quartile, Q2, is the overall median. Q1 and Q3 bound the middle half of the ordered distribution, with about 25% of values at or below Q1 and about 75% at or below Q3. State the quartile convention when using technology; these examples use medians of the ordered halves.

The pth percentile is described by the percentage of ordered data less than or equal to that value. In this sample, five of eight waits are at or below 12 minutes, or 62.5%. The first and third quartiles are the 25th and 75th percentiles, respectively. Keep “less than or equal to” in the comparison, particularly when equal values occur.

Measure the Spread

Three common measures of variability are range, interquartile range, and standard deviation. The range is maximum minus minimum: 18−4=14 minutes. The IQR is Q3−Q1: 15−7=8 minutes. Both have the units of the original measurements.

The sample standard deviation measures a typical deviation from the sample mean. It can be calculated with or without technology:

The squared deviations for this example are 49,25,9,1,1,9,25,49, whose sum is 168. Divide by 7 to obtain the sample variance, $s^2=24$ square minutes, then take the square root: $s=\sqrt{24}\approx4.899$ minutes. Variance has squared units; standard deviation returns to the data’s units. Do not divide by n when the requested measure is the sample standard deviation.

For technology, enter all eight values and request one-variable statistics. Check n=8, mean=11, and the sample standard deviation, often labeled Sx or s. A separate output labeled σx may use a denominator n and is a different statistic. The list, request, and choice of output all matter; pressing a summary button alone does not identify the requested measure.

Change Units Carefully

Changing measurement units changes calculated statistic values. Multiplying minute values by 60 to convert to seconds multiplies means, medians, quartiles, range, IQR, and standard deviation by 60. Variance multiplies by 60 squared. Adding a fixed amount shifts centers and positions but not the differences that measure spread.

For a positive linear unit conversion $y=a+bx$, center and position values become a plus b times the original, while range, IQR, and standard deviation multiply by b. For Celsius to Fahrenheit, a=32 and b=1.8: an 11°C mean becomes 51.8°F, but a 4.899°C standard deviation becomes about 8.818°F, without adding 32. This distinction follows because spread describes differences, not absolute readings.

Two Potential Outlier Rules

One frequently used rule flags values more than 1.5×IQR below Q1 or above Q3. Its fences are $Q_1-1.5\,IQR$ and $Q_3+1.5\,IQR$. A value exactly at a fence is not beyond it. A second rule flags values more than two standard deviations below or above the mean. Again, the comparison is strict: exactly two standard deviations does not exceed two.

Different rules can flag different observations. An outlier flag is a reason to inspect context and recording, not permission to erase an inconvenient value. State the rule and show its thresholds. Neither rule here requires pretending that every distribution is Normal.

For the eight worked waits, IQR fences are −5 and 27 minutes. The mean ±2s thresholds are about 1.202 and 20.798 minutes. None of the eight waits exceeds either pair of thresholds.

Compare More Than One Summary

Compare two or more independent samples using center, variability, shape, and outliers together. A sample with a higher mean but similar median may contain a large upper value. Two samples with the same IQR can differ in total range. Use units, actual values, and relevant distribution features rather than declaring one sample “more variable” without naming the measure.

Choose Resistant or Sensitive Summaries

The median is a resistant, or robust, measure of center, and IQR is a resistant measure of variability: isolated extremes do not greatly, if at all, change them. Mean, range, and standard deviation are nonresistant: an extreme can change them substantially. The mean still has useful purposes, such as representing total amount per observation, so resistance is a reasoned choice rather than a command never to use means.

Worked Example: An Extreme Wait

The waits 2,4,6,8,10,12,14,40 minutes have median 9 and IQR 8 minutes, but mean 12 and sample standard deviation 12 minutes. Replace 40 by 70: median and IQR stay 9 and 8, while mean becomes 15.75, range becomes 68, and standard deviation becomes about 22.282. For a typical wait in this right-tailed sample, median and IQR describe the bulk more steadily. For staffing based on total waiting minutes, the mean may answer a relevant question. Explain the choice with the distribution and the decision it serves.

Independent Practice

Use the information in each question to make your own calculation, representation, or argument. Explain what your result means in its setting.

Original generated practice for the Fall 2026 course.

Order before summarizing

Data A: 3,4,9 minutes. Data B: 4,6,8,10,12,14,16,18 minutes.

Part a

Calculate both medians, then B’s mean, minimum, maximum, Q1, Q2, and Q3 using medians of halves. What percentage of B is less than or equal to 12?

Your reasoning

Worked answer and rubric

Read the formula and units

The sample 4,10,10,16 cm has n=4, mean 10 cm, minimum 4 cm, maximum 16 cm, Q1=7 cm, Q3=13 cm, and sum of squared deviations 72 square centimetres. Quartiles use medians of halves.

Part a

Calculate range, IQR, sample variance, and sample standard deviation. Explain the denominator and units.

Your reasoning

Worked answer and rubric

Convert positions and differences

A set of times has mean 3 minutes, median 2.5 minutes, IQR 1 minute, and sample standard deviation 0.8 minute.

Part a

Calculate these four summaries in seconds and explain why the standard deviation changes.

Your reasoning

Worked answer and rubric

Strictly beyond the threshold

For one data set, Q1=10, Q3=18. For another, mean=10 and standard deviation=3. Use the specified rule for each data set.

Part a

Under the IQR rule, classify −2,−2.1,30,30.1. Under the two-SD rule, classify 4,3.9,16,16.1. Show thresholds and both tails.

Your reasoning

Worked answer and rubric

Similar center, different spread

Two independent service-time samples; all summaries in minutes.

Independent sample	Median, min	IQR, min	Range, min	Shape
A	12	4	10	Approximately symmetric
B	13	9	28	Right-skewed with one high outlier

Part a

Compare center and variability and incorporate shape/outliers.

Your reasoning

Worked answer and rubric

Typical experience or total burden?

A waiting-time sample has a long right tail from a few exceptionally large waits.

Part a

Justify summaries for a typical customer’s wait and for average total waiting burden per customer. Identify which center/spread measures resist extremes.

Your reasoning

Worked answer and rubric

Summarize an independent packing sample

Eight independently sampled packing times, in minutes, are 1,2,3,4,5,6,7,24. Use the median-of-halves quartile convention. For a comparison, a separate independent sample has median 5 minutes, IQR 3 minutes, mean 5 minutes, sample SD 2 minutes, approximately symmetric shape, and no flagged outliers.

Part a

Calculate mean and median by hand, showing the sum and ordered middle positions. State the smallest and largest times and explain their meanings.

Your reasoning

Part b

Calculate Q1, Q2, Q3, range, and IQR. Interpret Q1/Q3 and the percentage of times less than or equal to 6.

Your reasoning

Part c

Calculate the sample variance and standard deviation using squared deviations from your mean. Define x_i, x-bar, n, and s in your calculation, and interpret the variance and SD units. Explain what your SD means as a typical deviation of sampled packing times from their 6.5-minute mean.

Your reasoning

Part d

Describe how to enter these observations into technology to check the mean/median/SD, and identify which SD output is the sample one if the display reports Sx≈7.348 and σx≈6.874. Explain the distinction.

Your reasoning

Part e

Convert mean, median, range, IQR, and SD to seconds. Explain how an added constant would differ from a unit multiplier for spread.

Your reasoning

Part f

Apply both outlier methods to the 24-minute time. Show both pairs of thresholds and state whether the result is a reason automatically to delete it.

Your reasoning

Part g

Compare the two independent samples using centers, spread, shape, and outliers. Do not base the comparison only on the means.

Your reasoning

Part h

If 24 is replaced by 40, calculate new mean and range, identify what happens to median and IQR, and explain the SD’s change. Justify a center/spread pair for describing a typical packing time.

Your reasoning

Worked answer and rubric