To find the sum of the squared deviations of the individual data items about the mean, we can follow these steps:

1. Calculate the Mean

First, find the mean (average) of the data set.

$\text{Mean} = \frac{\sum x_i}{n}$

where $x_i$ are the individual data items and $n$ is the number of data points.

2. Find the Squared Deviations

For each data item, subtract the mean and square the result:

$\text{Squared Deviation} = (x_i - \text{Mean})^2$

3. Sum the Squared Deviations

Finally, sum all the squared deviations.

Now, let's calculate these step by step.

Step 1: Calculate the Mean

Given the data set:

$\{6.5, 8.1, 7.5, 7.9, 7.0, 4.3, 5.6, 7.3, 5.2, 6.8, 7.3, 7.6, 6.7, 8.0, 4.7\}$

The sum of the data items is:

$6.5 + 8.1 + 7.5 + 7.9 + 7.0 + 4.3 + 5.6 + 7.3 + 5.2 + 6.8 + 7.3 + 7.6 + 6.7 + 8.0 + 4.7 = 100.5$

The number of data items ($n$) is 15.

So, the mean is:

$\text{Mean} = \frac{100.5}{15} = 6.7$

Step 2: Calculate the Squared Deviations

Now, calculate the squared deviation for each data item:

(6.5 - 6.7)^2 & = (-0.2)^2 = 0.04 \\ (8.1 - 6.7)^2 & = (1.4)^2 = 1.96 \\ (7.5 - 6.7)^2 & = (0.8)^2 = 0.64 \\ (7.9 - 6.7)^2 & = (1.2)^2 = 1.44 \\ (7.0 - 6.7)^2 & = (0.3)^2 = 0.09 \\ (4.3 - 6.7)^2 & = (-2.4)^2 = 5.76 \\ (5.6 - 6.7)^2 & = (-1.1)^2 = 1.21 \\ (7.3 - 6.7)^2 & = (0.6)^2 = 0.36 \\ (5.2 - 6.7)^2 & = (-1.5)^2 = 2.25 \\ (6.8 - 6.7)^2 & = (0.1)^2 = 0.01 \\ (7.3 - 6.7)^2 & = (0.6)^2 = 0.36 \\ (7.6 - 6.7)^2 & = (0.9)^2 = 0.81 \\ (6.7 - 6.7)^2 & = (0.0)^2 = 0.00 \\ (8.0 - 6.7)^2 & = (1.3)^2 = 1.69 \\ (4.7 - 6.7)^2 & = (-2.0)^2 = 4.00 \\ \end{align*}$$ ### Step 3: Sum the Squared Deviations Add all the squared deviations: $$0.04 + 1.96 + 0.64 + 1.44 + 0.09 + 5.76 + 1.21 + 0.36 + 2.25 + 0.01 + 0.36 + 0.81 + 0.00 + 1.69 + 4.00 = 20.62$$ ### Final Answer: The sum of the squared deviations of the individual data items about the mean is **20.62**. Would you like any more details or have any questions? --- Here are five related questions you might consider: 1. How do you calculate the variance of a data set? 2. What is the relationship between variance and standard deviation? 3. How does the sum of squared deviations relate to sample size? 4. How would this calculation change if the data set were a sample instead of a population? 5. What are the practical implications of a high sum of squared deviations? **Tip:** The sum of squared deviations is a fundamental step in calculating both variance and standard deviation, which are key measures of spread in statistics.