Statistics · Answer guide

Lesson 1 · Estimate a grouped mean

Use class marks and frequencies, explaining the grouping assumption.

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

   Answer: 18. Class marks are 5, 15, 25. Σfx = 180, Σf = 10, so estimated mean = 18.

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

   Answer: 15. Class marks are 5, 15, 25. Σfx = 150, Σf = 10, so estimated mean = 15.

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

   Answer: 21. Class marks are 5, 15, 25. Σfx = 210, Σf = 10, so estimated mean = 21.

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

Teaching note: Classes with different frequencies do not get equal weight. A grouped mean is generally an estimate of the raw-data mean. Open responses need discussion; they are not automatically marked.

Lesson 2 · Estimate a grouped mode

Identify the modal class before substituting frequencies.

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

   Answer: 65/4. The modal class is 10–20. Mode = 10 + [(8−3)/(2×8−3−5)]×10 = 65/4.

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

   Answer: 50/3. The modal class is 10–20. Mode = 10 + [(10−2)/(2×10−2−6)]×10 = 50/3.

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

   Answer: 50/3. The modal class is 10–20. Mode = 10 + [(12−4)/(2×12−4−8)]×10 = 50/3.

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.

Teaching note: f₀ and f₂ mean frequencies of the neighbouring classes, not their class marks. Irregular or tied peaks need careful interpretation. Open responses need discussion; they are not automatically marked.

Lesson 3 · Find the median class by cumulative frequency

Use the halfway position and interpolation within a class.

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

   Answer: 15. N = 10, N/2 = 5. The median class is 10–20; l = 10, cumulative frequency before it = 2, f = 6, h = 10. Median = 15.

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

   Answer: 14. N = 10, N/2 = 5. The median class is 10–20; l = 10, cumulative frequency before it = 3, f = 5, h = 10. Median = 14.

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

   Answer: 110/7. N = 10, N/2 = 5. The median class is 10–20; l = 10, cumulative frequency before it = 1, f = 7, h = 10. Median = 110/7.

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.

Teaching note: Use the cumulative frequency before the median class, not the cumulative total at its end. Use class boundaries consistently. Open responses need discussion; they are not automatically marked.

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.

   12. Class marks are 5, 15, 25. Σfx = 120, Σf = 10, so estimated mean = 12.

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

   15. Class marks are 5, 15, 25. Σfx = 150, Σf = 10, so estimated mean = 15.

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

   50/3. The modal class is 10–20. Mode = 10 + [(9−3)/(2×9−3−6)]×10 = 50/3.

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

   16. The modal class is 10–20. Mode = 10 + [(8−2)/(2×8−2−4)]×10 = 16.

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

   35/2. N = 16, N/2 = 8. The median class is 10–20; l = 10, cumulative frequency before it = 2, f = 8, h = 10. Median = 35/2.

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

   50/3. N = 18, N/2 = 9. The median class is 10–20; l = 10, cumulative frequency before it = 3, f = 9, h = 10. Median = 50/3.

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. Class marks are 5, 15, 25. Σfx = 110, Σf = 10, so estimated mean = 11.

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

   16. Class marks are 5, 15, 25. Σfx = 160, Σf = 10, so estimated mean = 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. The modal class is 10–20. Mode = 10 + [(16−6)/(2×16−6−10)]×10 = 65/4.

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

   65/4. The modal class is 10–20. Mode = 10 + [(14−4)/(2×14−4−8)]×10 = 65/4.

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

   16. N = 20, N/2 = 10. The median class is 10–20; l = 10, cumulative frequency before it = 4, f = 10, h = 10. Median = 16.

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

   50/3. N = 14, N/2 = 7. The median class is 10–20; l = 10, cumulative frequency before it = 1, f = 9, h = 10. Median = 50/3.