Understanding Variance Calculation from Sums

The question asks us to calculate the variance of a set of 10 observations given the sum of the observations and the sum of their squares.

We are given:

• Number of observations, \(n = 10\)
• Sum of observations, \(\displaystyle \sum_{i = 1}^{10}x_i = 110\)
• Sum of squares of observations, \(\displaystyle \sum_{i = 1}^{10}x_i^2 = 1540\)

Variance is a measure of how spread out the data points are from the mean. There are two common types of variance: population variance (\(\sigma^2\)) and sample variance (\(s^2\)). When the data represents the entire population, we use the population variance formula. When it's a sample from a larger population, we typically use the sample variance formula. The question does not specify if this is a sample or population, but based on the options provided, the population variance formula is likely expected.

The formula for population variance (\(\sigma^2\)) is:

\[ \sigma^2 = \frac{\sum x_i^2}{n} - \left(\frac{\sum x_i}{n}\right)^2 \]

This formula requires the mean of the data, which is calculated as \(\frac{\sum x_i}{n}\). Let's first calculate the mean.

Step-by-Step Variance Calculation

Step 1: Calculate the Mean (\(\bar{x}\))

The mean is the sum of observations divided by the number of observations.

\[ \bar{x} = \frac{\sum x_i}{n} = \frac{110}{10} = 11 \]

The mean of the observations is 11.

Step 2: Calculate the Variance (\(\sigma^2\))

Now, we use the formula for population variance:

\[ \sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2 \]

Substitute the given values:

\[ \sigma^2 = \frac{1540}{10} - (11)^2 \]

\[ \sigma^2 = 154 - 121 \]

\[ \sigma^2 = 33 \]

The calculated variance is 33.

Let's verify if using the sample variance formula \(s^2 = \frac{1}{n-1} \left( \sum x_i^2 - \frac{(\sum x_i)^2}{n} \right)\) yields any of the options:

\[ s^2 = \frac{1}{10-1} \left( 1540 - \frac{(110)^2}{10} \right) \]

\[ s^2 = \frac{1}{9} \left( 1540 - \frac{12100}{10} \right) \]

\[ s^2 = \frac{1}{9} \left( 1540 - 1210 \right) \]

\[ s^2 = \frac{1}{9} (330) = \frac{330}{9} \approx 36.67 \]

Since 33 is one of the options and the sample variance is not, the question is asking for the population variance.

The variance of the given observations is 33.

Revision Table: Key Statistical Concepts

Additional Information on Variance and Standard Deviation

Variance and standard deviation are fundamental measures in statistics used to quantify the dispersion or spread of a set of data points around their mean. A higher variance or standard deviation indicates that the data points are more spread out from the mean, while a lower value indicates that they are clustered closer to the mean.

• Units: Variance is measured in units squared (e.g., if the data is in meters, variance is in square meters). Standard deviation is measured in the same units as the original data, making it easier to interpret in the context of the data.
• Interpretation: Standard deviation is often preferred for interpreting the spread because it is in the original units. For example, if the average test score is 75 and the standard deviation is 10, it suggests that typical scores are roughly within 10 points of the mean.
• Use Cases: Variance is used in many statistical tests and models (like ANOVA, regression) because its properties are mathematically convenient. Standard deviation is widely used in descriptive statistics, quality control, and risk assessment.
• Degrees of Freedom (n-1): The sample variance formula uses \(n-1\) in the denominator instead of \(n\). This is known as Bessel's correction and is used to provide a less biased estimate of the population variance when working with a sample. However, when calculating variance for a complete dataset (population), \(n\) is used.