1. Meaning of Variance
Variance measures how far the values in a data set spread out from the mean. It looks at the squared differences between each value and the mean, then finds the average of those squared differences.
If the variance is large, the values are widely scattered. If the variance is small, the values lie close to the mean.
Variance uses squared deviations, which helps give more weight to larger differences.
2. Formula for Variance
The formula for variance for raw (ungrouped) data is:
\( \sigma^2 = \dfrac{\sum (x - \bar{x})^2}{n} \)
Here:
- \( x \) = each data value
- \( \bar{x} \) = mean of the data
- \( (x - \bar{x})^2 \) = squared deviation
- \( n \) = number of values
Variance is always non-negative because squared values cannot be negative.
3. Meaning of Standard Deviation
Standard deviation is the square root of variance. It brings the measure of spread back to the same units as the original data, making it easier to understand and compare.
While variance tells the spread in terms of squared units, standard deviation tells the spread in the original units.
4. Formula for Standard Deviation
The formula for standard deviation is:
\( \sigma = \sqrt{\dfrac{\sum (x - \bar{x})^2}{n}} \)
This gives a measure of how much, on average, each value differs from the mean.
5. Shortcut Formula
Variance can also be calculated using the shortcut formula, which can sometimes make calculations simpler:
\( \sigma^2 = \dfrac{\sum x^2}{n} - \bar{x}^2 \)
Here:
- \( \sum x^2 \) = sum of squares of the data values
- \( \bar{x}^2 \) = square of the mean
6. Example
Consider the data: 2, 4, 6
Step 1: Calculate the mean
\( \bar{x} = \dfrac{2 + 4 + 6}{3} = 4 \)
Step 2: Find squared deviations
- (2 − 4)² = 4
- (4 − 4)² = 0
- (6 − 4)² = 4
Step 3: Sum of squared deviations
\( 4 + 0 + 4 = 8 \)
Step 4: Variance
\( \sigma^2 = \dfrac{8}{3} \approx 2.67 \)
Step 5: Standard Deviation
\( \sigma = \sqrt{\dfrac{8}{3}} \approx 1.63 \)
So, the variance is approximately 2.67 and the standard deviation is approximately 1.63.