Statistics · Learner pages

Lesson 1 · Estimate a grouped mean

Use class marks and frequencies, explaining the grouping assumption.

Name or initials: ____________   Date: ____________

The arithmetic mean is total value divided by the number of observations.

1. Replace each class by a representative. For classes 0–10, 10–20 and 20–30, the class marks are 5, 15 and 25.
2. Attach the frequencies. Suppose frequencies are 2, 3 and 5. The total number of observations is 10.
3. Build the weighted total. Σfx = 2×5 + 3×15 + 5×25 = 10 + 45 + 125 = 180.
4. Divide and state the limit. Estimated mean = 180/10 = 18. Grouped data hide exact individual values, so the result uses the class-mark approximation.

Worked example: For class marks 10, 20, 30 with frequencies 1, 2, 1, weighted total is 10 + 40 + 30 = 80 and total frequency is 4. The estimated mean is 20.

1. Grouped classes 0–10, 10–20, 20–30 have frequencies 2, 3, 5. Using class marks, find the estimated mean.

   1. 18 · 2. 20 · 3. 16

2. Grouped classes 0–10, 10–20, 20–30 have frequencies 3, 4, 3. Using class marks, find the estimated mean.

   1. 17 · 2. 13 · 3. 15

3. Grouped classes 0–10, 10–20, 20–30 have frequencies 1, 2, 7. Using class marks, find the estimated mean.

   1. 19 · 2. 21 · 3. 23

Discuss: Create two grouped distributions with the same classes but different frequencies. Explain how moving frequency toward the high class changes the estimated mean.

Lesson 2 · Estimate a grouped mode

Identify the modal class before substituting frequencies.

Name or initials: ____________   Date: ____________

The modal class has the greatest frequency; its midpoint need not be the estimated mode.

1. Locate the highest frequency. Classes 0–10, 10–20, 20–30 have frequencies 3, 8, 5. Thus 10–20 is the modal class.
2. Assign symbols by position. l = 10, h = 10, f₁ = 8, f₀ = 3 and f₂ = 5.
3. Substitute carefully. Mode = 10 + [(8 − 3)/(16 − 3 − 5)] × 10 = 10 + 50/8.
4. Interpret the estimate. Estimated mode = 16.25, which lies in 10–20. This estimates a concentration within the class, not an exact known raw observation.

Worked example: For neighbouring frequencies 4, 12, 8 around modal class 20–30, l = 20 and h = 10. Mode = 20 + (8/12)×10 = 80/3, about 26.67.

1. Classes 0–10, 10–20, 20–30 have frequencies 3, 8, 5. Find the estimated mode (an exact fraction is acceptable).

   1. 65/4 · 2. 73/4 · 3. 57/4

2. Classes 0–10, 10–20, 20–30 have frequencies 2, 10, 6. Find the estimated mode (an exact fraction is acceptable).

   1. 56/3 · 2. 44/3 · 3. 50/3

3. Classes 0–10, 10–20, 20–30 have frequencies 4, 12, 8. Find the estimated mode (an exact fraction is acceptable).

   1. 44/3 · 2. 50/3 · 3. 56/3

Discuss: Hold the peak frequency fixed and increase the next-class frequency. Use the formula to explore how the mode shifts toward that next class.

Lesson 3 · Find the median class by cumulative frequency

Use the halfway position and interpolation within a class.

Name or initials: ____________   Date: ____________

The median divides ordered observations into lower and upper halves.

1. Build running totals. For classes 0–10, 10–20, 20–30 with frequencies 2, 6, 2, the cumulative frequencies are 2, 8, 10.
2. Locate the halfway position. N = 10, so N/2 = 5. The running total passes 5 in the 10–20 class.
3. Substitute the correct values. l = 10, cf = 2, f = 6 and h = 10. Median = 10 + [(5 − 2)/6] × 10.
4. Interpret the interpolation. Estimated median = 15. The formula locates the halfway position within the class using a grouped-data approximation.

Worked example: For 0–10, 10–20, 20–30 with frequencies 3, 5, 2, N/2 = 5. The median class is 10–20. Median = 10 + ((5−3)/5)×10 = 14.

1. Classes 0–10, 10–20, 20–30 have frequencies 2, 6, 2. Find the estimated median.

   1. 15 · 2. 17 · 3. 13

2. Classes 0–10, 10–20, 20–30 have frequencies 3, 5, 2. Find the estimated median.

   1. 16 · 2. 12 · 3. 14

3. Classes 0–10, 10–20, 20–30 have frequencies 1, 7, 2. Find the estimated median.

   1. 96/7 · 2. 110/7 · 3. 124/7

Discuss: Make a frequency table and a cumulative-frequency column. Explain why identifying only the largest frequency would find a mode class, not a median class.

Final check · Set 1

Try one set after the lessons. Use the other set after reviewing any errors.

1. Grouped classes 0–10, 10–20, 20–30 have frequencies 5, 3, 2. Using class marks, find the estimated mean.

   14 / 10 / 12

2. Grouped classes 0–10, 10–20, 20–30 have frequencies 2, 6, 2. Using class marks, find the estimated mean.

   13 / 15 / 17

3. Classes 0–10, 10–20, 20–30 have frequencies 3, 9, 6. Find the estimated mode (an exact fraction is acceptable).

   56/3 / 44/3 / 50/3

4. Classes 0–10, 10–20, 20–30 have frequencies 2, 8, 4. Find the estimated mode (an exact fraction is acceptable).

   14 / 16 / 18

5. Classes 0–10, 10–20, 20–30 have frequencies 2, 8, 6. Find the estimated median.

   39/2 / 31/2 / 35/2

6. Classes 0–10, 10–20, 20–30 have frequencies 3, 9, 6. Find the estimated median.

   44/3 / 50/3 / 56/3

Final check · Set 2

Try one set after the lessons. Use the other set after reviewing any errors.

1. Grouped classes 0–10, 10–20, 20–30 have frequencies 6, 2, 2. Using class marks, find the estimated mean.

   11 / 13 / 9

2. Grouped classes 0–10, 10–20, 20–30 have frequencies 1, 7, 2. Using class marks, find the estimated mean.

   18 / 14 / 16

3. Classes 0–10, 10–20, 20–30 have frequencies 6, 16, 10. Find the estimated mode (an exact fraction is acceptable).

   65/4 / 73/4 / 57/4

4. Classes 0–10, 10–20, 20–30 have frequencies 4, 14, 8. Find the estimated mode (an exact fraction is acceptable).

   73/4 / 57/4 / 65/4

5. Classes 0–10, 10–20, 20–30 have frequencies 4, 10, 6. Find the estimated median.

   16 / 18 / 14

6. Classes 0–10, 10–20, 20–30 have frequencies 1, 9, 4. Find the estimated median.

   56/3 / 44/3 / 50/3