To solve the problem step-by-step, let's follow these instructions for each statistical measure: mean, median, mode, and midrange.
### Step 1: Calculate the Mean
The mean is the average of all data points. To find the mean, we use the formula:
[tex]\[
\text{Mean} = \frac{\sum (x_i \times f_i)}{N}
\][/tex]
where [tex]\(x_i\)[/tex] is the value of miles per gallon (MPG), [tex]\(f_i\)[/tex] is the frequency of [tex]\(x_i\)[/tex], and [tex]\(N\)[/tex] is the total number of cars (sum of all frequencies).
First, compute the total number of cars:
[tex]\[
N = 1 + 3 + 4 + 6 + 5 + 8 + 8 + 6 + 4 + 5 + 5 + 1 = 56
\][/tex]
Next, compute the sum of the products of each MPG and its frequency:
[tex]\[
\sum (x_i \times f_i) = (25 \times 1) + (26 \times 3) + (27 \times 4) + (28 \times 6) + (29 \times 5) + (30 \times 8) + (31 \times 8) + (32 \times 6) + (33 \times 4) + (34 \times 5) + (35 \times 5) + (36 \times 1)
\][/tex]
[tex]\[
= 25 + 78 + 108 + 168 + 145 + 240 + 248 + 192 + 132 + 170 + 175 + 36 = 1717
\][/tex]
Finally, calculate the mean:
[tex]\[
\text{Mean} = \frac{1717}{56} \approx 30.7
\][/tex]
So, the mean is approximately [tex]\(30.7\)[/tex].
### Step 2: Calculate the Median
The median is the middle value when the data set is ordered. Since we have 56 cars (an even number), the median will be the average of the 28th and 29th data points when all 56 data points are listed in order.
First, construct a cumulative frequency table to find the positions of the 28th and 29th data points:
[tex]\[
\begin{array}{c|c|c}
\text{MPG} & \text{Frequency} & \text{Cumulative Frequency} \\
\hline
25 & 1 & 1 \\
26 & 3 & 4 \\
27 & 4 & 8 \\
28 & 6 & 14 \\
29 & 5 & 19 \\
30 & 8 & 27 \\
31 & 8 & 35 \\
32 & 6 & 41 \\
33 & 4 & 45 \\
34 & 5 & 50 \\
35 & 5 & 55 \\
36 & 1 & 56 \\
\end{array}
\][/tex]
From the cumulative frequency, we see that the 28th and 29th data points both fall within the [tex]\(31\)[/tex] MPG category.
So, the median is [tex]\(31\)[/tex].
### Step 3: Calculate the Mode
The mode is the value that appears most frequently. By looking at the frequencies, we notice that both 30 and 31 have the highest frequency of [tex]\(8\)[/tex].
So, the mode is [tex]\(30\)[/tex] (since it's the smallest value in the case of a tie).
### Step 4: Calculate the Midrange
The midrange is the average of the maximum and minimum values in the data set.
[tex]\[
\text{Midrange} = \frac{\text{Max value} + \text{Min value}}{2} = \frac{36 + 25}{2} = \frac{61}{2} = 30.5
\][/tex]
So, the midrange is [tex]\(30.5\)[/tex].
### Summary
The final statistical values are:
- Mean: 30.7
- Median: 31
- Mode: 30
- Midrange: 30.5
