Worked example · Maths · Year 11
DCF: Data and computational thinking → Data and information literacy
A complete, secure Year 11 Higher tier response, annotated so you can see which success criterion each part meets. It is also the answer key. It practises GCSE Mathematics and Numeracy (Double Award) Unit 3: Calculator-allowed, sections 4.2.18 and 4.2.19 (cumulative frequency and box-and-whisker diagrams), 4.2.16 and 4.2.23 (comparing distributions and drawing conclusions), and 4.1.2 and 4.1.4 (testing a hypothesis and the reliability of data).
Key answers: cumulative frequencies 10, 22, 36, 52, 63, 71, 77, 80. From the curve: lower quartile about 18.5 Mbps (accept 17 to 20), median about 32.5 Mbps (accept 31 to 34), upper quartile about 47 Mbps (accept 46 to 48.5), interquartile range about 28.5 Mbps (accept 26 to 31). About 16 homes at 15 Mbps or less (accept 15 to 17), which is 20%.
(a) “Faster”: compare the medians, the typical speed. “More consistent”: compare the interquartile ranges, because a smaller spread means more similar speeds. I will use the interquartile range rather than the range, because the range depends only on the single slowest and fastest homes.
(b) Bryn Eira: 80 ÷ 214 × 100 = 37.4%. Cwm Derwen: 80 ÷ 297 × 100 = 26.9%.
Choosing the statistic before calculating shows the pupil has planned how the data will test each part of the hypothesis. Evidences criterion 3.
| Download speed, s (Mbps) | Working | Cumulative frequency |
|---|---|---|
| s ≤ 10 | 10 | 10 |
| s ≤ 20 | 10 + 12 | 22 |
| s ≤ 30 | 22 + 14 | 36 |
| s ≤ 40 | 36 + 16 | 52 |
| s ≤ 50 | 52 + 11 | 63 |
| s ≤ 60 | 63 + 8 | 71 |
| s ≤ 70 | 71 + 6 | 77 |
| s ≤ 80 | 77 + 3 | 80 |
The last cumulative frequency equals the number of homes, 80, which checks the running total. Evidences criterion 1.
Points plotted at the upper end of each class, starting at (0, 0), joined with a smooth curve. Reading lines drawn across from 20, 40 and 60, and up from 15 Mbps.
4(a)
| Lower quartile | Median | Upper quartile | Interquartile range |
|---|---|---|---|
| 18.5 Mbps | 32.5 Mbps | 47 Mbps | 47 − 18.5 = 28.5 Mbps |
4(b) Reading up from 15 Mbps meets the curve at about 16 homes. 16 ÷ 80 × 100 = 20% of the homes.
4(c)
| More than 30 Mbps | 10 Mbps or less | |
|---|---|---|
| Bryn Eira | 80 − 36 = 44 homes, 44 ÷ 80 = 55% | 10 homes, 10 ÷ 80 = 12.5% |
| Cwm Derwen | 80 − 16 = 64 homes, 64 ÷ 80 = 80% | 0 homes, 0% |
The reading lines are drawn on the graph, so the estimates can be checked. 4(c) uses the exact class boundaries at 10 and 30, so no estimate is needed there. Evidences criterion 1.
Bryn Eira: slowest 2.8, lower quartile 18.5, median 32.5, upper quartile 47, fastest 76.4 (Mbps).
Both box plots are on the same scale, so they can be compared directly. The slowest and fastest values come from the raw data, because they cannot be read from a cumulative frequency curve. Evidences criterion 2.
(a) The median download speed in Cwm Derwen (39.2 Mbps) is higher than in Bryn Eira (about 32.5 Mbps), so a typical home in Cwm Derwen gets a speed about 7 Mbps faster. The interquartile range in Cwm Derwen (47.5 − 31.5 = 16.0 Mbps) is much smaller than in Bryn Eira (about 28.5 Mbps), so speeds in Cwm Derwen are more consistent: the middle half of homes get similar speeds, while in Bryn Eira the speed you get depends a lot on which home you live in.
(b) Yes, the evidence supports both parts of the hypothesis for a typical home. The median is higher in Cwm Derwen and the interquartile range is just over half the size (16.0 compared with about 28.5 Mbps). 80% of Cwm Derwen homes got superfast speeds compared with 55% in Bryn Eira, and no Cwm Derwen home got 10 Mbps or less. However, it is not true for every home: the upper quartiles are almost the same (47.5 and 47 Mbps), and the fastest home in the whole survey (76.4 Mbps) was in Bryn Eira. This fits the background facts: Bryn Eira homes near the cabinet can be fast, but homes far up the valley are slow.
(c) The claim is wrong. 44 of the 80 Bryn Eira homes (55%) got more than 30 Mbps, and a quarter of homes got about 47 Mbps or more. But the resident has a point for some neighbours: 10 homes (12.5%) got 10 Mbps or less, which is below the Universal Service Obligation’s “decent” download speed.
(d)
| Why the results might not be reliable | How to make the survey fairer | |
|---|---|---|
| 1 | The homes chose themselves. People with slow broadband may be more likely to volunteer, so the sample may not represent the village. Only 37.4% of Bryn Eira and 26.9% of Cwm Derwen homes took part. | Choose a random sample from a list of every address, or a stratified sample with the same proportion of homes from each part of the valley. |
| 2 | Each home ran one test, at 20:00 to 21:00 on one evening, a busy time for home internet. One test can be unusually slow or fast. | Ask each home to run tests at several times across a week and use the median of their results. |
| 3 | 61% of Bryn Eira tests were over Wi-Fi (49 of 80), compared with 42.5% in Cwm Derwen (34 of 80). Wi-Fi usually gives lower results, so this is not a fair comparison. | Ask everyone to test with a cable, or compare Wi-Fi tests with Wi-Fi tests and cable tests with cable tests. |
(e) Using only the cable tests, the medians are 36.2 Mbps in Bryn Eira and 39.4 Mbps in Cwm Derwen, a difference of 3.2 Mbps. That is about half of the difference of about 7 Mbps when all the tests are used. So part of the gap came from how the tests were run, not from the broadband itself.
Modified conclusion: “Download speeds in Cwm Derwen are a little faster than in Bryn Eira for a typical home, and much more consistent. About half of the difference in typical speed in this survey came from more Bryn Eira homes testing over Wi-Fi. The real problem in Bryn Eira is the spread: 1 home in 8 gets 10 Mbps or less.”
New hypothesis to test next: “In Bryn Eira, homes further from the cabinet get slower download speeds, even when the test is run with a cable.”
(a) and (b) compare one average and one spread with figures and in context, and reach a judgement with a limit on it. (c) tests a claim with figures instead of opinion. (d) links each weakness to a specific fix. (e) uses new evidence to change the conclusion and write a sharper hypothesis. Evidences criteria 2, 3, 4 and 5.
Estimated mean. Midpoints 5, 15, 25, 35, 45, 55, 65, 75. Σfx = 50 + 180 + 350 + 560 + 495 + 440 + 390 + 225 = 2,690. Estimated mean = 2,690 ÷ 80 = 33.6 Mbps (the mean of the raw data is 34.0 Mbps). The median is the better average to use here, because it goes with the quartiles on the box plot and is not pulled by the very slowest and fastest homes. The mean and median are close, so the data are not strongly skewed.
Cwm Derwen curve. Plotting (0, 0), (10, 0), (20, 3), (30, 16), (40, 42), (50, 66), (60, 76), (70, 80), (80, 80) gives a median of about 39 Mbps and quartiles of about 31.5 and 47.5 Mbps, which agrees with the council.
A better survey. Record each home’s distance from the cabinet along the road. Choose homes by stratified random sampling from distance bands. Ask for three cable tests at different times. Draw a scatter diagram of speed against distance and look for negative correlation, remembering that correlation alone does not prove the distance causes the slow speed.
Cable tests only, from the CSV. Bryn Eira (31 homes): slowest 9.0, median 36.2, fastest 76.4, quartiles about 24 and 52 Mbps. Cwm Derwen (46 homes): slowest 11.2, median 39.4, fastest 66.8, quartiles about 33 and 48 Mbps. The medians move closer together, but Bryn Eira’s interquartile range (about 28 Mbps) is still almost double Cwm Derwen’s (about 15 Mbps). So “more consistent” still holds, and “faster” is weaker. Exact quartiles depend on the method or software used.