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Difficulty: MediumFrequency Distributions, Tables, and Grouped Data

The table below shows the frequency distribution of daily passenger counts (in hundreds) for a city bus route recorded over a 40-day period.

Daily Passengers (in hundreds)Frequency (Number of Days)
10−1410 - 1455
15−1915 - 191212
20−2420 - 241515
25−2925 - 2988

If the mean of the grouped data is estimated by using the midpoint of each class interval, what is the estimated mean daily passenger count (in hundreds)?

Answer: 20.25 hundred passengers

Answer

The estimated mean daily passenger count is 20.25 hundred passengers.
To calculate the estimated mean of grouped data, find the midpoint of each interval, multiply each midpoint by its frequency, sum those products (810810), and divide by the total number of observations (4040). This gives 81040=20.25\frac{810}{40} = 20.25.

Step-by-Step Solution

1
Calculate the midpoints for each of the four class intervals.
The midpoints are 10+142=12\frac{10+14}{2} = 12, 15+192=17\frac{15+19}{2} = 17, 20+242=22\frac{20+24}{2} = 22, and 25+292=27\frac{25+29}{2} = 27.
To estimate the mean of grouped frequency data, each interval is represented by its center value (midpoint).
2
Multiply each class midpoint by its frequency and calculate the total sum of these products.
(12×5)+(17×12)+(22×15)+(27×8)=60+204+330+216=810(12 \times 5) + (17 \times 12) + (22 \times 15) + (27 \times 8) = 60 + 204 + 330 + 216 = 810.
Multiplying each midpoint by its frequency computes the total estimated value contributed by all observations in that class.
3
Divide the total estimated value by the total sample size (total frequency).
81040=20.25\frac{810}{40} = 20.25.
The weighted average (grouped mean) is the sum of weighted midpoints divided by the total frequency.

Key Concept

Estimated Mean of Grouped Data
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Topics

Frequency Distributions, Tables, and Grouped Data

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