The highway speeds of 100 cars are summarized in the frequency distribution below. Find the mean speed.

Speed​ (mph)
Cars
​30-39
6
​40-49
16
​50-59
53
​60-69
15
​70-79
10

To find the mean speed from a frequency distribution, we follow these steps:

1. **Determine the midpoint for each class**: The midpoint (also called class mark) is the average of the lower and upper bounds of each class.
   
2. **Multiply the midpoint by the frequency**: This gives the total speed contribution of that class.

3. **Sum these products**: The total speed contribution from all classes.

4. **Divide by the total number of cars**: This gives the mean speed.

### Step-by-Step Calculation:

| Speed (mph) | Frequency (Cars) | Midpoint (Class Mark) | \( f \times x \) (Product) |
|:-----------:|:----------------:|:---------------------:|:--------------------------:|
|   30-39     |        6          |       \( \frac{30+39}{2} = 34.5 \)      |     \( 6 \times 34.5 = 207 \)     |
|   40-49     |       16          |       \( \frac{40+49}{2} = 44.5 \)      |     \( 16 \times 44.5 = 712 \)    |
|   50-59     |       53          |       \( \frac{50+59}{2} = 54.5 \)      |     \( 53 \times 54.5 = 2888.5 \) |
|   60-69     |       15          |       \( \frac{60+69}{2} = 64.5 \)      |     \( 15 \times 64.5 = 967.5 \)  |
|   70-79     |       10          |       \( \frac{70+79}{2} = 74.5 \)      |     \( 10 \times 74.5 = 745 \)    |

### Step 1: Calculate the total number of cars:
\[
\text{Total frequency} = 6 + 16 + 53 + 15 + 10 = 100
\]

### Step 2: Calculate the total of \( f \times x \):
\[
\text{Total product} = 207 + 712 + 2888.5 + 967.5 + 745 = 5520
\]

### Step 3: Calculate the mean speed:
\[
\text{Mean} = \frac{\text{Total of } f \times x}{\text{Total frequency}} = \frac{5520}{100} = 55.2 \text{ mph}
\]

### Conclusion:
The mean speed is **55.2 mph**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How does the mean speed compare to the mode in a frequency distribution?
2. What is the range of speeds observed in the data?
3. How do we calculate the median speed for this data?
4. What is the variance and standard deviation of these speeds?
5. How would a change in one of the frequencies affect the mean speed?

**Tip**: The midpoint of a class gives a reasonable estimate for the average value of all data points within that class.

The highway speeds of 100 cars are summarized in the frequency distribution below. Find the mean speed.

Speed​ (mph)  |  Cars
30-39        |  6
40-49        |  16
50-59        |  53
60-69        |  15
70-79        |  10

Learn how to calculate the mean speed of 100 cars using a frequency distribution table. This problem demonstrates key statistical concepts like frequency distribution, midpoints, and measures of central tendency, with a step-by-step calculation resulting in a mean speed of 55.2 mph.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation