Connecting the Dots... · Lesson 6 of 13
Zero vs. No value
“Choose the right observations and denominator by separating genuine zeros from missing data.”
• Distinguish a recorded zero from a missing entry. • Calculate mean and median using the relevant observations. • Explain the denominator for per-appearance comparisons. • Compare full-year and explicitly selected seasonal data. • Recognise limits caused by missing survey responses.
Zero vs. No value
A zero is a number with a definite meaning: the observation was made, and the measured count was none. A missing entry means something else: the observation may not have been obtained or the event did not take place. Before computing a mean or median, decide what each entry represents and which question the dataset should answer.
| Entry | Meaning | Treatment for runs per appearance |
|---|---|---|
| 0 runs | The player appeared and made no runs. | Include zero and count the appearance. |
| Did not play | There was no appearance in that match. | Do not invent a zero; exclude it from an appearance-based count. |
| Unknown result | The result was not recorded or is unavailable. | Keep it identified as missing and explain the limitation. |
Problem
A player’s entries are 57, 13, 0, 84, —, 51, 27. Find mean runs per appearance.
- 1.Six entries record actual appearances, including the 0. The dash denotes a non-appearance.
- 2.Total = 57 + 13 + 0 + 84 + 51 + 27 = 232 runs.
- 3.Mean = 232 ÷ 6 ≈ 38.67 runs per appearance. Dividing by 7 answers a different scheduled-match question.
Deleting a genuine zero would raise the mean incorrectly: 232 ÷ 5 = 46.4 no longer describes all appearances. Replacing a missing entry with zero would also change the question and may introduce a false observation. The same distinction matters for survey responses: no pets is a valid response; an absent student has not yet provided a response.
Zero Median Runs Scored!
A team can have a large total while most individual players contribute very little. The source’s innings has eleven individual run totals and nineteen extras. To investigate the runs made by players, the extras must be separated from those observations.
| Player | Runs |
|---|---|
| Zak Crawley | 19 |
| Ben Duckett | 0 |
| Ollie Pope | 0 |
| Joe Root | 22 |
| Harry Brook | 158 |
| Ben Stokes | 0 |
| Jamie Smith | 184 |
| Chris Woakes | 5 |
| Brydon Carse | 0 |
| Josh Tongue | 0 |
| Shoaib Bashir | 0 |
Problem
How can the players’ median be zero when the team total is 407?
- 1.Six of the eleven players have 0. In sorted order, the first six entries are therefore 0.
- 2.The middle of eleven values is the sixth, so median individual runs = 0.
- 3.The player totals sum to 388. The 19 extras bring the team total to 407.
- 4.Mean individual runs = (407 − 19) ÷ 11 = 388 ÷ 11 ≈ 35.27. The mean and median highlight different features of this uneven contribution.
Choosing the observations deliberately
Sita’s tree produces monthly mango counts 0, 0, 8, 24, 41, 16, 5, 0, 0, 0, 0, 0. If we ask about average production across a full year, every month belongs in the data. If we ask about an explicitly defined fruiting period, the selected months answer a different question.
| Question | Included observations | Mean mangoes per month | Median |
|---|---|---|---|
| Across the full year | All 12 months | 94 ÷ 12 ≈ 7.83 | 0 |
| During March–July | 8, 24, 41, 16, 5 | 94 ÷ 5 = 18.8 | 16 |
Problem
Why do the annual and March–July mango means differ?
- 1.Both totals are 8 + 24 + 41 + 16 + 5 = 94 mangoes.
- 2.The full-year mean divides by 12; the March–July mean divides by 5.
- 3.The seasonal mean describes only the stated selected period. It must not be labelled the average for the entire year.
Never drop zeros simply because they reduce an average. Select observations according to a clearly stated question, and identify missing entries separately. When selecting a season, name the months rather than silently keeping only positive counts.
A survey with absent students
Sanskruti records household pet and domestic-animal counts. Two students are absent, while several present students report zero. We can calculate centres for the twenty recorded responses, but we cannot pretend those responses include the absent students.
The entries are 0, 1, 0, 4, 8, 0, 0, 2, 1, 1, 5, 3, 4, 0, 0, —, 10, 25, 2, —, 2, 4. There are 22 listed students but only 20 known responses. The high value 25 stands apart from most responses and has a large influence on the total.
Problem
Find mean and median for the pet survey.
- 1.Keep the six genuine zeros. Exclude the two missing entries only from this calculation on known responses.
- 2.There are 20 responses with total 72. Mean = 72 ÷ 20 = 3.6 animals per responding household.
- 3.Sorted responses: 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 5, 8, 10, 25. The tenth and eleventh entries are 2, so median = 2.
- 4.The value 25 is high relative to most responses. These centres describe the respondents; the full-class centres remain unknown until the missing data is collected.
Quiz
A recorded zero in a count should generally be:
A player appears in six matches, including one with zero runs. For runs per appearance, divide by:
What does an absent student’s dash mean in a pet survey?
Six of eleven players have zero runs. The median is:
For average mango production per month across a whole year, use:
Practice Problems
- Find mean runs per appearance for 8, 0, 12, and “did not play”.
- Why is discarding the zero from 8, 0, 12 misleading?
- Player A has 14, 16, 10, 10; B has 0, 8, 6, 4; C has 8, 11, “did not play”, 13. Compare mean points per appearance.
- Find the yearly and March–July median of Sita’s mango counts.
- A survey has responses 0, 2, 4 and two absent people. What mean can you report?
- In the innings example, why must 19 extras be subtracted before finding mean individual player runs?
There are 3 appearances, not 4. Total = 20; mean = 20 ÷ 3 ≈ 6.67 runs per appearance.
Key Takeaways
• Zero and missing data have different meanings. • Include genuine zeros when their events belong in the question. • Match the denominator to the observations being described. • Missing responses limit what can be said about the whole group. • Clearly label averages calculated for a selected season or subgroup. • A large total can coexist with a zero median in uneven data.