Question

the gas mileages (in miles per gallon) for 33 cars are shown in the frequency distribution. approximate the mean of the frequency distribution.
gas mileage (in miles per gallon) frequency
30 - 35 12
36 - 41 11
42 - 47 3
48 - 53 7
the approximate mean of the frequency distribution is
(round to one decimal place as needed.)

Explanation:

Step1: Find the mid - points

For the interval 30 - 35, the mid - point $x_1=\frac{30 + 35}{2}=32.5$.
For the interval 36 - 41, the mid - point $x_2=\frac{36+41}{2}=38.5$.
For the interval 42 - 47, the mid - point $x_3=\frac{42 + 47}{2}=44.5$.
For the interval 48 - 53, the mid - point $x_4=\frac{48+53}{2}=50.5$.

Step2: Calculate the product of mid - points and frequencies

$f_1 = 12$, $f_1x_1=12\times32.5 = 390$.
$f_2 = 11$, $f_2x_2=11\times38.5 = 423.5$.
$f_3 = 3$, $f_3x_3=3\times44.5 = 133.5$.
$f_4 = 7$, $f_4x_4=7\times50.5 = 353.5$.

Step3: Find the sum of products and sum of frequencies

$\sum_{i = 1}^{4}f_ix_i=390 + 423.5+133.5 + 353.5=1300.5$.
$\sum_{i = 1}^{4}f_i=12 + 11+3 + 7=33$.

Step4: Calculate the mean

The mean $\bar{x}=\frac{\sum_{i = 1}^{4}f_ix_i}{\sum_{i = 1}^{4}f_i}=\frac{1300.5}{33}\approx39.4$.

Answer:

39.4