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A-Level

Statistics: Standard Deviation & Variance

Learn standard deviation and variance step by step β€” what they measure, how to calculate them, and the 68-95-99.7 rule for normal distributions.

You already know averages (mean, median, mode). But averages only tell you the middle.Standard deviation tells you how spread out the data is. Is everyone clustered around the average, or all over the place?

Quick Mean Refresher

Mean (xΜ„) = Sum of all values Γ· Number of values

What Is Variance?

Variance measures how far each data point is from the mean, on average. Here's how to calculate it:

  1. Find the mean
  2. Subtract the mean from each value (these are deviations)
  3. Square each deviation (to remove negatives)
  4. Find the mean of the squared deviations

Worked Example 1 β€” Calculating Variance

Data: 4, 8, 6, 5, 7

  1. Mean = (4+8+6+5+7) Γ· 5 = 30 Γ· 5 = 6
  2. Deviations: βˆ’2, 2, 0, βˆ’1, 1
  3. Squared: 4, 4, 0, 1, 1
  4. Variance = (4+4+0+1+1) Γ· 5 = 10 Γ· 5 = 2

Standard Deviation

Standard Deviation = √Variance

From our example: SD = √2 β‰ˆ 1.41

A small standard deviation means the data is tightly clustered around the mean. A large one means it's widely spread.

Worked Example 2 β€” Comparing Two Data Sets

Test scores for two classes (both with mean = 70):

  • Class A: 68, 69, 70, 71, 72 β†’ SD β‰ˆ 1.4 (very consistent)
  • Class B: 50, 60, 70, 80, 90 β†’ SD β‰ˆ 14.1 (big range of ability)

Same average, but very different spreads. Standard deviation reveals this.

The 68-95-99.7 Rule (Normal Distribution)

For data that follows a bell curve (normal distribution):

Range% of data
Within 1 standard deviation of mean68%
Within 2 standard deviations95%
Within 3 standard deviations99.7%

Example: if exam scores have mean 60 and SD 10, then about 68% of students scored between 50 and 70.

Worked Example 3 β€” Using the Rule

Heights of adults: mean = 170 cm, SD = 8 cm.

  • 68% are between 162 cm and 178 cm
  • 95% are between 154 cm and 186 cm
  • 99.7% are between 146 cm and 194 cm

Someone 195 cm tall would be taller than 99.85% of people β€” extremely rare!

Try It Yourself

  1. Find the variance of: 10, 12, 14, 16, 18. (Answer: mean=14, variance=8)
  2. If SD = 5, what is the variance? (Answer: 25)
  3. Weights have mean 75 kg, SD 6 kg. What range covers 95%? (Answer: 63 kg to 87 kg)

Calculate standard deviation for your own data set.

β†’ Open the Standard Deviation Calculator