Range and Standard Deviation

Measures of central tendency (mean, median, mode) tell you where the center of your data is, but they do not tell you how spread out the data is. Two datasets can have the same mean but look completely different. Range and standard deviation measure this spread — how much individual values differ from each other and from the center.

Range

The range is the simplest measure of spread. It is the difference between the highest and lowest values in a dataset.

$\text{Range} = \text{Maximum value} - \text{Minimum value}$

Example 1: Daily High Temperatures

Daily high temperatures for one week (in $\degree$F): 72, 68, 75, 80, 71, 69, 77.

$\text{Range} = 80 - 68 = 12\degree\text{F}$

Answer: The range is 12$\degree$F, meaning daily highs varied by 12 degrees over the week.

The range is easy to calculate, but it has a major weakness: it only uses two values (the highest and lowest). A single outlier can make the range misleading.

Example 2: Range with an Outlier

Test scores: 82, 85, 88, 84, 86, 45.

$\text{Range} = 88 - 45 = 43$

Without the outlier (45), the range would be $88 - 82 = 6$. One unusually low score makes the range suggest far more variability than the majority of the data actually has. This is why we need standard deviation.

Variance and Standard Deviation

Standard deviation measures how far values typically fall from the mean. It uses every data point — not just the extremes.

The calculation has two stages: first you find the variance, then you take its square root.

Formulas

Variance (for a population):

$\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$

Standard deviation (for a population):

$\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}$

Where:

• $x_i$ = each individual data value
• $\mu$ = the mean of the dataset
• $N$ = the number of data values
• $\sum$ = “the sum of”

For a sample (a subset of a larger population), divide by $N - 1$ instead of $N$. On most standardized tests and in introductory courses, you will use the population formula above.

Example 3: Step-by-Step Standard Deviation

Dataset: 4, 8, 6, 5, 7

Step 1: Find the mean.

$\mu = \frac{4 + 8 + 6 + 5 + 7}{5} = \frac{30}{5} = 6$

Step 2: Subtract the mean from each value and square the result.

Step 3: Find the variance (average of squared deviations).

$\sigma^2 = \frac{4 + 4 + 0 + 1 + 1}{5} = \frac{10}{5} = 2$

Step 4: Take the square root to get the standard deviation.

$\sigma = \sqrt{2} \approx 1.41$

Answer: The standard deviation is approximately 1.41.

This tells you that values in this dataset typically fall about 1.41 units away from the mean of 6.

What Standard Deviation Tells You

• Low SD = data points are clustered close to the mean (consistent, reliable).
• High SD = data points are spread far from the mean (variable, inconsistent).

Consider two students’ test scores, both with a mean of 80:

• Student A: 78, 80, 82, 79, 81 (SD $\approx$ 1.4) — very consistent
• Student B: 65, 95, 70, 90, 80 (SD $\approx$ 11.4) — highly variable

Same average, very different stories. Standard deviation reveals the difference.

Quick Reference

Real-World Application: Electrician — Measuring Voltage Consistency

An electrician tests the voltage output of a circuit at five different times throughout the day to check for stability. The expected voltage is 120 V.

Readings (in volts): 119.5, 120.2, 120.0, 119.8, 120.5

Step 1: Find the mean.

$\mu = \frac{119.5 + 120.2 + 120.0 + 119.8 + 120.5}{5} = \frac{600.0}{5} = 120.0 \text{ V}$

Step 2: Find the squared deviations.

Step 3: Calculate variance and standard deviation.

$\sigma^2 = \frac{0.25 + 0.04 + 0.00 + 0.04 + 0.25}{5} = \frac{0.58}{5} = 0.116$

$\sigma = \sqrt{0.116} \approx 0.34 \text{ V}$

Interpretation: With a standard deviation of only 0.34 V, this circuit delivers very stable power. Voltage fluctuations are minimal and well within the acceptable range. If the SD were much higher — say 5 V or more — the electrician would investigate for loose connections, overloaded circuits, or faulty equipment.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Range ($\text{Max} - \text{Min}$) is quick to calculate but easily distorted by a single outlier.
• Standard deviation measures how far values typically fall from the mean — it uses every data point.
• A low standard deviation means data is consistent and tightly clustered. A high standard deviation means data is spread out and variable.
• To calculate SD: find the mean, subtract it from each value, square the deviations, average them (variance), and take the square root.
• Two datasets can share the same mean but have very different standard deviations — always check the spread.

Return to Statistics for more topics in this section.