CH-12-001 · Histogram frequencies, proportion and estimated mean

The histogram shows journey times for a group of students. The frequency densities for the intervals 0–10, 10–30 and 30–60 minutes are respectively 9, 3, 5.

(a) Estimate the mean journey time. Give your answer to 2 decimal places.

(b) Estimate the proportion of students whose journey takes less than 20 minutes. State the assumption used.

Part (a) Histograms ("cow food"), Estimated Mean & Linear Interpolation Booklet p. 67 & p. 72

Method

1) Use f = c.w × f.d to find frequencies

2) Use midpoint (m) and fm table for Mean = Total fm ÷ Total f

3) Interpolate within class assuming uniform distribution

Worked Example (CH-12-001 · Part (a))

Estimate the mean journey time. Give your answer to 2 decimal places.

Booklet p. 72:

\text{Frequency Density (f.d)} = \frac{\text{Frequency (f)}}{\text{Class Width (c.w)}}

1) Multiply each density by class width.

2) Frequencies 90, 60, 150.

3) Use midpoints 5, 20, 45 and divide the weighted sum by 300.

4) 28.00 minutes

28.00 minutes

Part (b) Histograms ("cow food"), Estimated Mean & Linear Interpolation Booklet p. 67 & p. 72

Method

1) Use f = c.w × f.d to find frequencies

2) Use midpoint (m) and fm table for Mean = Total fm ÷ Total f

3) Interpolate within class assuming uniform distribution

Worked Example (CH-12-001 · Part (b))

Estimate the proportion of students whose journey takes less than 20 minutes. State the assumption used.

Booklet p. 72:

\text{Frequency Density (f.d)} = \frac{\text{Frequency (f)}}{\text{Class Width (c.w)}}

1) Use 90 + half of 60, then divide by 300.

\frac{2}{5}

3) Assume a uniform distribution within the 10–30 minute interval.

\frac{2}{5}

; assume times are uniformly distributed within 10–30 minutes.