Range, Variance, and Standard Deviation
In the previous lesson, we learned how to describe the center of data using mean, median, and mode. However, knowing only the center is not enough.
Two datasets can have the same average but behave very differently. To understand how data is spread out, we use measures of dispersion.
The three most important measures of dispersion are: Range, Variance, and Standard Deviation.
What Is Range?
The range is the simplest measure of spread. It is calculated as the difference between the highest and lowest values in a dataset.
Numerical Example
Consider the following data:
10, 20, 30, 40, 50
Highest value = 50
Lowest value = 10
Range = 50 − 10 = 40
The range gives a quick idea of how wide the data is, but it is very sensitive to extreme values.
Real-World Example (Range)
If the highest temperature in a week is 38°C and the lowest is 22°C, the range of temperatures is 16°C.
Limitations of Range
Although range is easy to calculate, it has an important drawback:
- It depends only on two values
- It ignores how data is distributed in between
Because of this, we use more reliable measures such as variance and standard deviation.
What Is Variance?
The variance measures how far each data value is from the mean. It considers all data points, not just the extremes.
In simple words, variance tells us how much the data varies around the average.
Conceptual Explanation
To calculate variance, we:
- Find the mean
- Find the difference between each value and the mean
- Square those differences
- Take the average of the squared differences
We square the differences to avoid positive and negative values canceling each other.
Simple Numerical Example
Data:
2, 4, 6
Mean = (2 + 4 + 6) ÷ 3 = 4
Differences from mean:
- 2 − 4 = −2
- 4 − 4 = 0
- 6 − 4 = 2
Squared differences:
- (−2)² = 4
- 0² = 0
- 2² = 4
Variance = (4 + 0 + 4) ÷ 3 = 2.67
Variance is expressed in squared units, which can be hard to interpret.
What Is Standard Deviation?
The standard deviation is simply the square root of the variance.
It brings the measure of spread back to the same unit as the original data, making it much easier to understand.
Continuing the Example
Variance = 2.67
Standard Deviation = √2.67 ≈ 1.63
This means that, on average, data values are about 1.63 units away from the mean.
Real-World Example (Standard Deviation)
In finance, standard deviation is used to measure risk.
If two investments have the same average return, the one with a higher standard deviation is considered more risky because its returns vary more.
Comparison of Measures
| Measure | What It Shows | Main Limitation |
|---|---|---|
| Range | Overall spread using min and max | Ignores most data points |
| Variance | Average squared deviation from mean | Hard to interpret units |
| Standard Deviation | Typical distance from the mean | Requires more calculation |
Quick Check
Dataset A: 10, 10, 10, 10
Dataset B: 5, 10, 15, 20
Both datasets have the same mean. Which dataset has a larger spread?
Practice Quiz
Question 1:
Which measure of dispersion is most affected by extreme values?
Question 2:
Why is standard deviation preferred over variance?
Question 3:
If all values in a dataset are the same, what is the standard deviation?
Mini Practice
A class has the following test scores:
70, 70, 70, 70
- What is the range?
- Would the standard deviation be high or low?
What’s Next
In the next lesson, we will study Percentiles and Quartiles, which help us understand the relative position of data values.