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8B

The 68–95–99.7 rule

The empirical rule for normal distributions: about 68%, 95% and 99.7% of data lie within 1, 2 and 3 standard deviations of the mean, and using the symmetric pieces to find percentages.

Here is the payoff for learning standard deviation in Year 11. If data is normal, you don't need the full dataset to know how it's spread. The mean and standard deviation alone tell you what percentage of the data lies in any band of whole standard deviations.

The rule

For any normal distribution, approximately:

  • $68\%$ of the data lies within $1$ standard deviation of the mean: between $\bar{x} - s$ and $\bar{x} + s$.
  • $95\%$ lies within $2$ standard deviations: between $\bar{x} - 2s$ and $\bar{x} + 2s$.
  • $99.7\%$ lies within $3$ standard deviations: between $\bar{x} - 3s$ and $\bar{x} + 3s$.

This is on the HSC reference sheet, but you need to use it fast, so learn it. It's also called the empirical rule.

Example 1●

The heights of Year 12 boys at a school are normally distributed with mean $176$ cm and standard deviation $7$ cm. Find the interval that contains the middle $95\%$ of heights.

$2s = 2 \times 7 = 14$
95% is within 2 standard deviations, so find 2s first.
$176 - 14 = 162$ and $176 + 14 = 190$
Go 2s below and 2s above the mean.
About $95\%$ of the boys are between $162$ cm and $190$ cm tall.
State the answer in context.

Breaking the curve into pieces

Because the curve is symmetric, each band splits evenly in half. Work out every piece from the three big numbers:

RegionWorkingPercentage
Mean to $1s$ (one side)$68 \div 2$$34\%$
$1s$ to $2s$ (one side)$(95 - 68) \div 2$$13.5\%$
$2s$ to $3s$ (one side)$(99.7 - 95) \div 2$$2.35\%$
Beyond $3s$ (one side)$(100 - 99.7) \div 2$$0.15\%$
Above the meanhalf the data$50\%$
Example 2●●

For the same heights (mean $176$, $s = 7$), what percentage of boys are (a) taller than $190$ cm, (b) between $169$ cm and $190$ cm?

$190 = 176 + 2 \times 7$, so $190$ is $2s$ above the mean, and $169 = 176 - 7$ is $1s$ below.
Label the curve: 155, 162, 169, 176, 183, 190, 197.
(a) $(100 - 95) \div 2 = 2.5\%$
Above 2s is half of what's outside 95%.
(b) $34 + 34 + 13.5 = 81.5\%$
169 to 176 is one piece of 34%. 176 to 190 is 34% plus 13.5%.
InteractiveShade the bell curve
Try this

Use the sliders to set the mean and standard deviation, then press 68%, 95% or 99.7% to shade that region. The interval appears underneath.

Change the standard deviation. Does the shaded percentage change? What does change? Then try Pick a band: tap two markers and see the percentage between them.

Your turn

The time Year 12 students take to complete a 5 km fun run is normally distributed with mean $32$ minutes and standard deviation $4$ minutes. What percentage of students take between $24$ and $36$ minutes? If $600$ students ran, how many is that?

Show the answer

$24 = 32 - 2 \times 4$ is $2s$ below, and $36 = 32 + 4$ is $1s$ above. So the percentage is $13.5 + 34 + 34 = 81.5\%$. Then $0.815 \times 600 = 489$ students.

Flashcards
The 68–95–99.7 rule
About $68\%$, $95\%$ and $99.7\%$ of normal data lie within $1$, $2$ and $3$ standard deviations of the mean.
The pieces on one side
$34\%$, $13.5\%$, $2.35\%$, $0.15\%$ (mean to $1s$, $1s$ to $2s$, $2s$ to $3s$, beyond $3s$).
Percentage more than two standard deviations above the mean
$2.5\%$, which is half of the $5\%$ outside the middle $95\%$.
Practice

  1. For normal data, about what percentage lies within $2$ standard deviations of the mean?

  2. Marks are normal with mean $60$ and $s = 10$. About $68\%$ of marks lie between:

  3. What percentage of normal data lies more than $1$ standard deviation above the mean?

  4. Marks are normal with mean $60$ and $s = 10$. What percentage of marks are below $40$?

  5. For normal data, what percentage lies between the mean and $2$ standard deviations above it?

  6. Bags of chips have masses normally distributed with mean $170$ g and $s = 5$ g. What percentage of bags weigh between $160$ g and $175$ g?

  7. Data is normal with mean $50$. About $99.7\%$ of values lie between $35$ and $65$. The standard deviation is:

  8. What percentage of normal data lies more than $3$ standard deviations below the mean?

  9. Exam style: The lifetimes of a brand of LED globe are normal with mean $25{,}000$ hours and $s = 2{,}000$ hours. What percentage of globes last more than $23{,}000$ hours?

  10. Exam style: At a Newcastle school, $1{,}080$ students sat a test. Marks were normal with mean $64$ and $s = 9$. How many students scored above $82$?

Formula sheet