Find the mean for the data items in the given frequency distribution.
​Score, x
1
2
3
4
5
6
7
8
 
​Frequency, f
2
1
5
6
4
5
6
2


To find the mean of a frequency distribution, you can use the following formula:

\[
\text{Mean} = \frac{\sum (x \cdot f)}{\sum f}
\]

Where:
- \(x\) represents the score.
- \(f\) represents the frequency of each score.

Let's calculate the mean step by step:

1. Multiply each score by its corresponding frequency to get \(x \cdot f\).
2. Sum all the \(x \cdot f\) values.
3. Sum all the frequencies.
4. Divide the sum of \(x \cdot f\) by the sum of the frequencies.

### Step 1: Calculate \(x \cdot f\)
\[
\begin{array}{c|c|c}
\text{Score } (x) & \text{Frequency } (f) & x \cdot f \\
\hline
1 & 2 & 1 \cdot 2 = 2 \\
2 & 1 & 2 \cdot 1 = 2 \\
3 & 5 & 3 \cdot 5 = 15 \\
4 & 6 & 4 \cdot 6 = 24 \\
5 & 4 & 5 \cdot 4 = 20 \\
6 & 5 & 6 \cdot 5 = 30 \\
7 & 6 & 7 \cdot 6 = 42 \\
8 & 2 & 8 \cdot 2 = 16 \\
\end{array}
\]

### Step 2: Sum of \(x \cdot f\)
\[
\sum (x \cdot f) = 2 + 2 + 15 + 24 + 20 + 30 + 42 + 16 = 151
\]

### Step 3: Sum of frequencies
\[
\sum f = 2 + 1 + 5 + 6 + 4 + 5 + 6 + 2 = 31
\]

### Step 4: Calculate the mean
\[
\text{Mean} = \frac{151}{31} \approx 4.87
\]

So, the mean of the data items in the given frequency distribution is approximately **4.87**.

Would you like more details or have any questions?

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Here are some related questions to further explore this topic:

1. What is the median of the same frequency distribution?
2. How do you calculate the mode for this frequency distribution?
3. What is the range of the data set?
4. How would the mean change if the score of 7 had a frequency of 8?
5. Can you calculate the variance for this frequency distribution?
6. How would you compute the standard deviation for this data?
7. What happens to the mean if all frequencies are doubled?
8. How does removing the highest and lowest scores affect the mean?

**Tip:** When dealing with frequency distributions, always double-check the sum of the frequencies to ensure accuracy before calculating the mean.

Learn how to calculate the mean from a frequency distribution with an example problem. Step-by-step solution provided using the formula Mean = (Σ(x * f)) / Σf. Suitable for high school students studying statistics.

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