The arithmetic mean of the following frequency distribution of number of accidents Xon week working days is:X: 2 4 6 8 10 12 Frequency: 3 4 2 1 4 2
6.625
The question asks for the arithmetic mean of a given frequency distribution. The frequency distribution shows the number of accidents (X) on week working days and the corresponding frequency for each number of accidents.
Calculating the Arithmetic Mean of a Frequency Distribution
The arithmetic mean for a frequency distribution is calculated using the formula:
$$ \bar{X} = \frac{\sum (f \times X)}{\sum f} $$
Where:
- $\bar{X}$ is the arithmetic mean.
- $X$ represents the value or class midpoint (in this case, the number of accidents).
- $f$ represents the frequency of each value.
- $\sum (f \times X)$ is the sum of the products of each value and its frequency.
- $\sum f$ is the sum of all frequencies (total number of observations).
Step-by-Step Arithmetic Mean Calculation
Let's organize the data and calculate the necessary sums. We have the number of accidents (X) and their frequencies (f):
| Number of Accidents (X) | Frequency (f) | f × X |
|---|---|---|
| 2 | 3 | 2 × 3 = 6 |
| 4 | 4 | 4 × 4 = 16 |
| 6 | 2 | 6 × 2 = 12 |
| 8 | 1 | 8 × 1 = 8 |
| 10 | 4 | 10 × 4 = 40 |
| 12 | 2 | 12 × 2 = 24 |
Now, we calculate the sum of frequencies ($\sum f$) and the sum of the products ($\sum (f \times X)$):
- Sum of frequencies ($\sum f$): $$ 3 + 4 + 2 + 1 + 4 + 2 = 16 $$
- Sum of products ($\sum (f \times X)$): $$ 6 + 16 + 12 + 8 + 40 + 24 = 106 $$
Finally, apply the formula for the arithmetic mean:
$$ \bar{X} = \frac{\sum (f \times X)}{\sum f} = \frac{106}{16} $$
Performing the division:
$$ \bar{X} = 6.625 $$
The arithmetic mean of the given frequency distribution is 6.625.
Revision Table: Key Statistics Concepts
| Concept | Definition | Calculation Method |
|---|---|---|
| Arithmetic Mean | A measure of central tendency; the average of a dataset. | Sum of values divided by the number of values (for raw data) or $$\frac{\sum (f \times X)}{\sum f}$$ (for frequency distribution). |
| Frequency Distribution | A table or graph that shows the frequency of different outcomes in a sample. | Organizes data into classes or categories and counts how many observations fall into each. |
| Summation ($\sum$) | The operation of adding a sequence of numbers. | Indicated by the Greek capital letter sigma. |
Additional Information: Understanding Frequency Distributions
A frequency distribution is a way to summarize data. It tells us how often each value or range of values appears in a dataset. In this problem, the frequency distribution shows how many working days had a specific number of accidents.
The arithmetic mean calculated from a frequency distribution is essentially a weighted average, where each value of X is weighted by its frequency. This is because the formula takes into account how many times each X value occurs in the dataset.
Understanding frequency distributions is fundamental in statistics as they are used to calculate various measures like the mean, median, mode, variance, and standard deviation, providing insights into the data's central tendency, spread, and shape.
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