Descriptive Statistics Classroom Activity

The hook, 5 minutes

Give students the table below. Every record is invented for this activity; it describes no real workers, customers, or orders. One row represents one packing order. Station is the label of the fictional station that packed it. Time is elapsed packing time in minutes, recorded to the nearest minute. Item count is the number of items in the order.

Ask students to predict which station has the larger typical packing time and which has more variable times. Require a reason pointing to the data before calculating. Then ask: does averaging the order IDs tell us anything about packing performance? Save the explanations for the check below.

Guided exploration on the calculator, 25 minutes

Use the written table rather than the calculator presets. Paste each station's six packing times separately into the descriptive statistics calculator, or calculate on paper. Record the station label with every output. When drawing the two dotplots or boxplots, use a common horizontal scale from 0 to 20 minutes.

Student worksheet, part 1: what was recorded? Name the observational unit and give the number of units overall and in each group. Classify Order ID, Station, Item count, and Packing time. Explain why a whole-number recording does not necessarily make the underlying variable discrete.

Part 2: compare the original groups. For each station, find the sum, mean, median, minimum, maximum, quartiles, IQR, range, and both 1.5 IQR fences. Identify flagged observations. Use the median-excluded convention: sort the list, split by ordered position into two equal halves, and exclude only the single middle observation when the total count is odd. Values equal to the median remain wherever their positions put them.

Draw the distributions on the common scale. Write a comparison that connects both groups in each sentence: center in minutes, variability in minutes, and shape or unusual features visible in the data. Does this invented comparison establish that using Station B causes slower packing? Why or why not?

Part 3: predict, then change one value. In a separate hypothetical copy, change Order 112 from 18 to 8 minutes. Do not delete it. Predict which of Station B's mean, median, IQR, and range will change, then recompute all four. Is this hypothetical exercise a justification for changing the original record? Explain.

Use about 5 minutes for part 1, 12 for part 2, and 8 for part 3. The worked answers below let students check the arithmetic after committing to their predictions.

Check for understanding, 5 minutes

Use these questions in pairs before revealing the key.

1. What is the difference between the unit, the variable Packing time, and the entry 18?
2. Why are Order ID and Item count not the same kind of numerical information?
3. Complete a direct comparison: Station B's median packing time is ___ minutes ___ than Station A's, and its IQR is ___ times Station A's.
4. What is the correct conclusion about Order 112 from the outlier rule alone?

Teacher answer key: (1) The unit is one order; Packing time is a characteristic measured on each order; 18 minutes is its value for Order 112. There are 12 units, six per station. (2) Order ID is a label, not a quantitative measurement; its numerical size has no packing interpretation. Item count is a discrete quantitative variable. Station is categorical. Elapsed time is quantitative and continuous even though recorded values are rounded to whole minutes. (3) Station B's median of 6.5 minutes is 1 minute higher than A's 5.5; B's IQR of 3 minutes is three times A's 1 minute. (4) Order 112 is flagged above B's 12.5-minute upper fence. The rule does not tell us whether it is wrong.

If students confuse the first two answers, use variables and data types practice. If their comparisons list two summaries without connecting the groups, use comparing distributions practice.

Exit ticket, 5 minutes

Answer individually using the same packing-order data.

1. After the hypothetical 18-to-8 change, has Station B become faster than Station A at the median? Support the answer with both values.
2. A student says, "The 18 is flagged, so replace it with 8 in the report." Is that justified?
3. Can we infer that the station caused the original time difference?

Teacher answer key: (1) No. B's median stays 6.5 minutes, compared with A's 5.5 minutes, so B is still 1 minute higher at the median. (2) No. We changed a value only to study sensitivity. An outlier flag is not evidence of a recording error; retain the original observation unless there is an independently supported reason to correct it. (3) No. These are invented descriptive records, not evidence from a randomized experiment. Even analogous real observational records could differ in order complexity or other conditions.

Collect the slips to distinguish a calculation problem from an interpretation problem. The session totals 40 minutes: 5 for the hook, 25 for exploration, 5 for the check, and 5 for the exit ticket.

No-tech variant

Run the same predictions and questions on paper. Give each pair the 12-row table and two blank five-number-summary templates. Make six time cards per station and arrange them on parallel 0-to-20-minute number lines. Keep repeated values as separate cards.

For the sensitivity exercise, replace B's 18 card with an 8 card and sort again. Leave the original table unchanged. Students can see that B's middle pair, 6 and 7, has not moved, even though its sum and maximum have changed. Have each pair explain one unchanged summary and one changed summary using positions or arithmetic, not just calculator output.

The same check and exit ticket work without devices. Keep the answer key hidden until students have written their comparisons.

Materials

• One copy of the invented packing-order table and worksheet questions per pair. No student grades or personal measurements are needed.
• Scratch paper for sorting, sums, quartiles, and fences.
• Two number lines with the same 0-to-20-minute scale, or a projector with the descriptive statistics calculator. Enter the activity data yourself; default presets are different data.
• For the paper version, six packing-time cards for each station and a spare 8-minute card.
• Exit ticket slips with the three questions above.

For a shorter period, move the sensitivity exercise to homework and use the first worked example to check the original comparison.

Common misconceptions this lesson targets

1. A number can be an identifier rather than a measurement. Order 112 is not "more order" than Order 101. Averaging their labels has no useful interpretation, while averaging item counts or packing times does.
2. Mean and median answer different questions. The original B mean is 8 minutes, but its median is 6.5 and five of its six times are between 4 and 8. The median is useful for a typical-order comparison here; the mean still matters if the question concerns total time per order. Do not discard a mean simply because an outlier exists.
3. Range and IQR use different parts of the ordered data. In the hypothetical change, B's range falls from 14 to 4 minutes, while its IQR stays 3. This example demonstrates resistance, not a promise that quartiles never change when observations change.
4. An outlier flag is not a diagnosis or a causal explanation. The fence identifies 18 as unusual relative to the rest of B. It neither establishes an error nor explains why that order took longer. Describe the observed groups without claiming the station caused the difference.