At a glance. One lesson of about 60 minutes, opening the unit. Students estimate a mean from mid-interval values, name the modal and median classes, find a missing frequency, and state limits for the true mean.
Calculator: GDC optional. The sum is short enough to do by hand. The optional section shows a list with a frequency list, and warns that the GDC cannot turn an estimate into an exact answer.
What students will learn
Know
- The estimated mean is Σfx ÷ n using mid-interval values.
- The modal class has the greatest frequency; the median class holds the middle value.
- The true mean lies between the lower-boundary mean and the upper-boundary mean.
Understand
- Grouping throws information away, so a statistic from a grouped table is an estimate with a stated error.
- A mean is a weighted average: a group with more values counts for more.
- Narrower classes give a smaller possible error.
Do
- Estimate a mean from a grouped table.
- Find a missing frequency from a given mean.
- State the least and greatest the true mean could be.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: estimating the mean, finding the median class and solving for a missing frequency (questions 1 to 7).
- Criterion B. Investigating patterns: the what-if. Students predict how the range of possible means changes with the class width and generalize.
- Criterion C. Communicating: setting out the table with mid-interval values, fx and the sum, and stating the answer as an estimate with its limits.
- Criterion D. Applying mathematics in real-life contexts: the clinic hook, where only classes are published and a decision still has to be made.
The one thing to do with the figure
Ask for the extremes before sliding. With the slider at 5, ask: “If I move every value to the lower end of its class, will the mean go up or down, and by how much?” Then slide to 0 and let students check against the readout.
Then ask why the least and greatest means are exactly one class width apart however the frequencies are shared out. Every value moves by the same amount, so the mean moves by that amount.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the clinic table. Collect predictions of the least and greatest mean. |
| 12 min | The figure with predictions; the key idea; settle the hook. |
| 10 min | Warning card and worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose C or D, return to the warning card and the weighting. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 16; 10; 25. |
|---|---|
| Hook | Not exactly: between the lower-boundary mean and the upper-boundary mean, best estimate halfway. |
| Worked examples | 3.8 kg; modal and median class both 2 to under 4 kg; between 2.8 kg and 4.8 kg. |
| Faded example | 9.25 minutes, within 2.5 minutes of the true mean. |
| 1. Estimated mean of a two-class table | B. A uses lower boundaries; C ignores the frequencies; D divides by the number of classes. |
| 2. Estimated mean from a table | 19. 570 ÷ 30. |
| 3. Lower boundary of the median class | 30. The 18th value is in 30 to under 40. |
| 4. Greatest the true mean can be | 31.4. 26.4 + 5. |
| 5. Missing frequency from a given mean | 4. 380 + 25f = 400 + 20f. |
| 6. Largest whole number the true mean can be | 45. 42.6 + 2.5 = 45.1. |
| 7. Combining two groups by weight | 44. 1760 ÷ 40. |
| 8. What a grouped table gives exactly | C. The frequencies are exact; A, B and D depend on positions inside classes. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Used lower boundaries, ignored the frequencies, or divided by the number of classes. | Make the four columns: class, mid-interval value, f, fx. |
| 14, 20 or 142.5 in question 2 | Used lower boundaries, ignored the frequencies, or divided by the number of classes. | Make the table: class, mid-interval value, f, fx; then n at the foot. |
| 20 or 10 in question 3 | Gave the modal class, or looked for the position 17.5 as if it were a value. | Running totals, then find the class that holds the 18th value. |
| 36.4 or 21.4 in question 4 | Added the whole width, or gave the least instead of the greatest. | Draw the number line: the estimate is the middle of a range of width w. |
| 0.8 or 5 in question 5 | Left f out of n, or gave a distance from the mean. | Write the whole fraction with f in the top and in the bottom. |
| 47 or 46 in question 6 | Added the whole width, or rounded up. | Greatest it can be is 45.1; the whole numbers at or below are ... 44, 45. |
| 40 or 1760 in question 7 | Averaged the means without weights, or forgot to divide. | Totals first: mean times number of values. |
| A, B or D in question 8 | Treated an estimate as exact. | For each option ask: which values would I need to know? |
Questions worth asking
- Can two different data sets give the same grouped table? Yes: any values that fall into the same classes. That is why only an estimate is possible.
- Why is the midpoint the best single guess for a class? It is the value that is nearest, on average, to values spread evenly across the class, and it limits the error to half a width.
- Is the estimate more likely to be above or below the true mean for skewed data? It can be either; if values bunch near the lower end of each class the estimate is too high.
- Why do we report the estimate with fewer decimal places than the calculator shows? It is only as accurate as the grouping allows; a long decimal claims precision the table does not have.
Differentiation
A way in for anyone stuck
- Give a four-column template: class, mid-interval value, frequency f, fx, with a total row.
- Write the mid-interval values first and have students check each is halfway between the boundaries.
- For the missing frequency, build the equation in words first: total of fx divided by total of f equals the mean.
Stretch
- Show that for equal class width w the mean using lower boundaries is always w less than the mean using upper boundaries.
- Decide, for a table with a very wide top class, whether the estimated mean can be trusted.
Students learning in English as an additional language
“Class” here means a group of values between two boundaries, not a school class. “Mid-interval” means halfway between. Say “x bar” for the mean and “sigma f x” for the total of frequency times value.
| Word | Plain meaning | Example |
|---|---|---|
| class | a group of values between two boundaries | 0 ≤ x < 8 |
| mid-interval value | the value halfway between the class boundaries | 6 for the class 4 ≤ x < 8 |
| modal class | the class with the greatest frequency (equal widths) | the tallest bar of a histogram |
| median class | the class that holds the middle value | the class where the running total passes n ÷ 2 |
| estimate | a value found from incomplete information | x̄ ≈ Σfx ÷ n |
| class width | the upper boundary minus the lower boundary | 8 − 4 = 4 |
Sentence starters for explaining aloud or in writing:
- The mid-interval value of this class is … because …
- The mean is an estimate because …
- The true mean must lie between … and …
Two things not to say
“The mean is Σfx divided by the number of classes.” The divisor is the number of values, n.
“The estimated mean is the real mean.” It is a good guess with a stated possible error.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- EstimateClasses 0 to under 4 (f = 3), 4 to under 8 (f = 5), 8 to under 12 (f = 2). Estimate the mean.
5.6. The mid-interval values are 2, 6 and 10, so Σfx = 6 + 30 + 20 = 56 and 56 ÷ 10 = 5.6. - ClassWhich class holds the median of 21 values when the running totals are 4, 13 and 21?
The second class: the 11th value is first reached there.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA market in Hanoi records stall rents in classes 200 wide. The estimated mean is 1 450. Between what limits is the true mean?
Within 100: between 1 350 and 1 550. - Earlier work (Year 9)A mean of 6 values is 7.5. A seventh value of 10.5 is added. What is the new mean?
7.9. The total is 45 + 10.5 = 55.5, and 55.5 ÷ 7 = 7.93, so 7.9 to one decimal place.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA grouped table has one class with frequency 0. Does removing that class change the estimated mean? Say why.
No: it adds 0 to Σfx and 0 to n. - ProofShow that if every value in a table is increased by k, the estimated mean increases by k.
Σf(x + k) ÷ n = Σfx ÷ n + kΣf ÷ n = x̄ + k.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to the slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
The With a GDC section is closed by default and optional. Students without a GDC lose nothing; every answer is reached by hand.