The following summary statistics are given for a data set:

   14,

61




Calculate the variance of the data (correct to 2dp).

To calculate the variance of a data set, we need more than just the two numbers "14" and "61." Typically, we need either the entire data set or the mean (average) and the number of data points (sample size). Given only "14" and "61," it's unclear whether these are the data points, a summary statistic like the mean and sum of squares, or something else.

Assuming these numbers are data points, I'll proceed to calculate the variance for the data set {14, 61}.

### Step-by-Step Calculation:

1. **Calculate the mean (average):**

\[
\text{Mean} = \frac{14 + 61}{2} = \frac{75}{2} = 37.5
\]

2. **Calculate the squared differences from the mean for each data point:**

\[
(14 - 37.5)^2 = (-23.5)^2 = 552.25
\]
\[
(61 - 37.5)^2 = (23.5)^2 = 552.25
\]

3. **Calculate the variance:**

Since we are dealing with a sample (only two data points), we'll use the sample variance formula:

\[
\text{Variance} = \frac{\sum (\text{Squared differences})}{n-1} = \frac{552.25 + 552.25}{2 - 1} = \frac{1104.5}{1} = 1104.5
\]

Thus, the variance of the data set is **1104.50** (rounded to two decimal places).

Would you like more details or have any questions? Here are some related questions to consider:

1. What is the difference between population variance and sample variance?
2. How does the number of data points affect the variance?
3. Can variance ever be negative?
4. What is the standard deviation, and how is it related to variance?
5. How would you calculate variance for a larger data set?
6. What is the impact of outliers on variance?
7. Why do we divide by \(n-1\) for sample variance?
8. Can variance be used to compare different data sets?

**Tip:** Remember that the variance gives you a sense of how much the data points deviate from the mean, with higher variance indicating greater spread.

Learn how to calculate the variance of a data set {14, 61} with a detailed step-by-step guide. Understand the sample variance formula and its application in statistics. Suitable for high school students and anyone interested in statistics.

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