Summary Statistics: Calculate and Choose — Worked Solutions
Worked Solutions
Compare your reasoning with the worked solutions after attempting the questions. A different valid representation or explanation can earn credit when it satisfies the stated criteria.
Original generated practice for the Fall 2026 course. Return to the lesson and independent checks.
Order before summarizing: worked answer
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?
A median=4. B median=11, mean=88/8=11, minimum=4, maximum=18, Q1=7, Q2=11, Q3=15. Five of eight B values are ≤12, so 62.5%. Quartile interpretations are about 25% at/below Q1, 75% at/below Q3, and middle half between Q1/Q3.
| Evidence for credit | Points |
|---|---|
| Odd/even medians and mean/min/max. | 1 |
| Correct quartiles with convention and positions. | 1 |
| Inclusive percentage 62.5%. | 1 |
Read the formula and units: worked answer
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.
Range=16−4=12 cm; IQR=13−7=6 cm; s²=72/(4−1)=24 cm²; s=√24≈4.899 cm. The sample formula divides squared deviations by n−1, then takes the square root.
| Evidence for credit | Points |
|---|---|
| Four correct measures. | 1 |
| Sample denominator and squared/original units distinguished. | 1 |
Convert positions and differences: worked answer
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.
Mean=180 s, median=150 s, IQR=60 s, and s=48 s. Every time and every time difference is multiplied by 60, so positions and spread change units consistently.
| Evidence for credit | Points |
|---|---|
| Four correct converted values and contextual difference explanation. | 1 |
Strictly beyond the threshold: worked answer
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.
IQR=8, fences −2 and 30: −2 and 30 are not beyond fences, while −2.1 and 30.1 are outliers by this rule. Mean±2SD gives 4 and 16: endpoints 4 and 16 are not beyond, while 3.9 and 16.1 are outliers.
| Evidence for credit | Points |
|---|---|
| Both threshold calculations. | 1 |
| All eight strict-boundary classifications and both tails. | 1 |
Similar center, different spread: worked answer
| 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.
B’s median is 1 minute higher, a modest center difference. B has larger IQR (9 vs 4) and range (28 vs 10), so greater spread by both measures, with right skew and a high outlier. A is more concentrated and approximately symmetric.
| Evidence for credit | Points |
|---|---|
| Numerical center and both spread comparisons. | 1 |
| Shape/outlier context included. | 1 |
Typical experience or total burden?: worked answer
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.
Median and IQR are suitable for typical/bulk experience because they resist extremes. Mean can describe total waiting minutes divided by customers, while standard deviation describes spread around it but is sensitive to large waits. Mean, range, and standard deviation are nonresistant; median and IQR are resistant.
| Evidence for credit | Points |
|---|---|
| Justifies choices using question and distribution. | 1 |
| All resistant/nonresistant categories correctly identified. | 1 |
Summarize an independent packing sample: worked answer
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.
Sum=52, mean=6.5 minutes. Median=(4+5)/2=4.5 minutes. Minimum=1 and maximum=24 minutes describe the smallest/largest sampled packing times.
| Evidence for credit | Points |
|---|---|
| Correct sum/mean and even median with steps. | 1 |
| Contextual extrema. | 1 |
Part b
Calculate Q1, Q2, Q3, range, and IQR. Interpret Q1/Q3 and the percentage of times less than or equal to 6.
Q1=2.5, Q2=4.5, Q3=6.5 minutes. Range=23 minutes and IQR=4 minutes. Quartiles describe about 25% and 75% at/below their positions and bound the middle half. Six of eight times are ≤6, so 75%; Q1/Q3 are the 25th/75th percentiles.
| Evidence for credit | Points |
|---|---|
| All positions and spreads with units. | 1 |
| Approximate quartile, middle-half, and inclusive percentage interpretations. | 1 |
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.
x_i is each sampled packing time, x-bar=6.5 minutes, n=8, and s is sample SD. Squared deviations sum to 378 min². s²=378/7=54 min²; s=√54≈7.348 minutes. The SD describes a typical deviation of these sampled packing times from their 6.5-minute mean; it does not promise that every time lies within 7.348 minutes of the mean. Variance is in squared minutes and SD is in minutes.
| Evidence for credit | Points |
|---|---|
| Correct definitions and squared-deviation sum. | 1 |
| n−1 division, variance, root, and units. | 1 |
| Interprets about 7.348 minutes as a typical deviation of sampled packing times from their 6.5-minute mean, without a guaranteed bound. | 1 |
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.
Enter all eight times into one list, request one-variable summaries, and check n=8, mean=6.5, and median=4.5. Sx≈7.348 is the sample SD because it uses n−1; σx≈6.874 uses n. Choosing the sample label and confirming the complete list are necessary to check the requested calculation.
| Evidence for credit | Points |
|---|---|
| Complete-list and count/summary procedure. | 1 |
| Correct sample output and denominator distinction. | 1 |
Part e
Convert mean, median, range, IQR, and SD to seconds. Explain how an added constant would differ from a unit multiplier for spread.
Multiply by 60: mean 390 s, median 270 s, range 1380 s, IQR 240 s, SD≈440.908 s. A fixed addition shifts positions and centers but cancels in differences, leaving range/IQR/SD unchanged.
| Evidence for credit | Points |
|---|---|
| All five unit conversions. | 1 |
| Difference between shifting positions and scaling spread. | 1 |
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.
IQR fences=2.5−6=−3.5 and 6.5+6=12.5 minutes; 24 exceeds the upper fence. Mean±2s≈−8.197 and 21.197 minutes; 24 also exceeds the upper threshold. It is flagged by both rules but should be checked in context, not automatically deleted.
| Evidence for credit | Points |
|---|---|
| Both threshold pairs and correct strict classifications. | 1 |
| No automatic deletion. | 1 |
Part g
Compare the two independent samples using centers, spread, shape, and outliers. Do not base the comparison only on the means.
The first median 4.5 is slightly below the second median 5, despite its larger mean 6.5 versus 5. The first IQR 4 and SD≈7.348 exceed 3 and 2; the high 24-minute outlier/right tail affect its mean and SD. The second is approximately symmetric with no flagged outliers, so the center comparison depends on the chosen summary.
| Evidence for credit | Points |
|---|---|
| Both centers, spread comparisons, and shape/outlier explanation. | 1 |
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.
New mean=68/8=8.5 minutes; range=39 minutes; median stays 4.5 and IQR stays 4. SD rises to √166≈12.884 minutes because squared deviations increase. Median/IQR are resistant and fit typical bulk experience; mean/range/SD are nonresistant. A different justified decision purpose can support a mean-based summary.
| Evidence for credit | Points |
|---|---|
| Correct changed and unchanged summaries. | 1 |
| SD sensitivity and all resistance categories. | 1 |
| Contextual justification rather than universal rule. | 1 |