Understanding Variance and Standard Deviation

In statistics, we use measures to describe the spread or dispersion of a dataset. Two fundamental measures of spread are variance and standard deviation. They are closely related, and understanding their relationship is key to solving this type of problem.

What is Variance?

Variance measures how far each number in the set is from the mean, and thus from every other number in the set. It is the average of the squared differences from the Mean.

What is Standard Deviation?

Standard deviation is the square root of the variance. It measures the typical distance of the data points from the mean. A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range of values.

Relationship Between Variance and Standard Deviation

The standard deviation is simply the positive square root of the variance. If $\sigma^2$ represents the variance and $\sigma$ represents the standard deviation, the relationship is:

$$ \sigma = \sqrt{\sigma^2} $$

We take the positive square root because the standard deviation represents a distance or a measure of spread, which cannot be negative. While the square root of a positive number technically has both a positive and a negative value (e.g., $\sqrt{4} = \pm 2$), in the context of standard deviation, we only consider the positive value.

Calculating the Standard Deviation from Variance

The question provides the variance of a set of data as 196. We need to find the standard deviation of the data.

Given:

• Variance ($\sigma^2$) = 196

We need to find the standard deviation ($\sigma$).

Using the relationship $\sigma = \sqrt{\sigma^2}$, we can calculate the standard deviation:

$$ \sigma = \sqrt{196} $$

To find $\sqrt{196}$, we need to find a number that, when multiplied by itself, equals 196.

• Let's test some numbers:
• $10 \times 10 = 100$ (Too small)
• $12 \times 12 = 144$ (Too small)
• $13 \times 13 = 169$ (Too small)
• $14 \times 14 = 196$ (Exactly 196)

So, the square root of 196 is 14.

$$ \sigma = 14 $$

The standard deviation must be a non-negative value, as it represents a measure of spread or distance. Therefore, the standard deviation is 14.

Analyzing the Options

Let's look at the given options based on our calculation:

• A. $\pm 14$: This includes the negative value, which is not used for standard deviation.
• B. 14: This is the positive square root, matching our calculation.
• C. 96: This is not the square root of 196.
• D. 98: This is not the square root of 196.

Our calculated standard deviation is 14, which corresponds to option B.

Revision Table: Key Statistical Measures

Additional Information on Variance and Standard Deviation

Understanding standard deviation and variance is crucial in many fields, including finance, quality control, and scientific research.

• Units: Variance is measured in units squared (e.g., if data is in meters, variance is in square meters). Standard deviation is measured in the same units as the original data (e.g., meters). This makes standard deviation easier to interpret than variance.
• Sample vs. Population: There are slight differences in the formulas for calculating variance and standard deviation for a sample versus a population. For a population, we divide by the total number of data points (N). For a sample, we typically divide by N-1 (Bessel's correction) to provide a less biased estimate of the population variance/standard deviation. However, the relationship $\sigma = \sqrt{\sigma^2}$ holds true regardless of whether it's for a population or a sample.
• Use Cases: Standard deviation is often used to compare the spread of different datasets or to understand the risk associated with an investment. Variance is used in various statistical tests and models (like ANOVA).