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Apply the 68–95–99.7% rule

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Ten observations are 8, 8, 9, 9, 10, 10, 11, 11, 12 and 12. Six lie between 8.51 and 11.49. What percentage is that?

Year 11, VCE Units 1 & 2, Chapter 2, 4 difficulty levels, Calculator allowedBuild a worksheet

Standard-deviation intervals

Observed percentages come from counts. For an approximately symmetric bell-shaped distribution, the empirical rule instead gives approximate model percentages

\[\bar{x}\pm ks\]
  1. Central bands Within one, two and three standard deviations: about 68%, 95% and 99.7% respectively
  2. Outside and one tail Subtract the central percentage from 100. Divide the remainder by two for one tail
  3. The other side For everything except one tail, subtract that tail from 100%
  4. Unequal distances Split the interval at the mean and add the two appropriate half-band percentages
  5. Estimated count Divide the estimated percentage by 100, then multiply by the total count

Worked example

Bell-shaped packing times have mean 50 seconds and standard deviation 5 seconds. Estimate the percentage taking less than 60 seconds

\[\bar{x}=50,\quad s=5\]
  1. \(\displaystyle (60-50)/5=2\)The threshold is two standard deviations above the mean
  2. \(\displaystyle 100-95=5\)About 5% lies outside the central two-standard-deviation band, across both tails
  3. \(\displaystyle 5/2=2.5\)The upper tail alone is about 2.5%
  4. \(\displaystyle 100-2.5=97.5\)The requested percentage is below the threshold: approximately 97.5%

Common mistakes

Using both tails for one side
✕\(\displaystyle P(X>\bar{x}+2s)\approx5\%\)✓\(\displaystyle P(X>\bar{x}+2s)\approx2.5\%\)
The outside 5% is split equally between the two tails
Forgetting the complement
✕\(\displaystyle P(X<\bar{x}+2s)\approx2.5\%\)✓\(\displaystyle P(X<\bar{x}+2s)\approx97.5\%\)
Below the upper threshold is the large region, not the small upper tail
Confusing count and percentage
✕\(\displaystyle N=81.5\)✓\(\displaystyle N=0.815\times2000=1630\)
81.5 is a percentage. Multiply its decimal form by the total to estimate a count

Difficulty levels

introIdentify the central 68% and 95% bands
standardUse all three central bands and find their endpoints
challengeCheck shape, use tails and complements, reverse a threshold, combine unequal bands and estimate counts
examSelect the matching central percentage or endpoint pair

Questions about this topic

Is symmetry enough?+

No. The empirical rule needs an approximately symmetric, bell-shaped distribution. Flat and two-peaked symmetric distributions need not follow it

Are the percentages exact?+

They are approximate model percentages. When told to use the 68–95–99.7% rule, calculate from those stated values. For an actual observed percentage, count the values inside the interval instead

What does central mean?+

The endpoints are equally far below and above the mean. A percentage alone does not identify an arbitrary interval unless its position is specified

Does below mean a small tail?+

Not necessarily. Below an upper threshold includes most of the distribution. Draw or identify the requested region before calculating

Do I need to calculate the standard deviation again?+

Not when it is supplied. Use it to locate endpoints. Calculating standard deviation from raw data is a separate skill taught in the previous section

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, General Mathematics, chapter 2: Data distributions

In this chapter

Classify a variable as nominal, ordinal, discrete or continuousDistinguish interval and ratio measurement scalesBuild a frequency and percentage frequency tableBuild a bar chart from a frequency table2A Data types and categorical displays: Read values from a frequency table or bar chart2B Reading categorical displays: Read values from a frequency table or bar chartReport on the distribution of a categorical variableCompare categorical distributions across two groupsGroup numerical data into class intervalsConstruct a histogram from a frequency tableRead values from a histogramDescribe the shape of a distributionConstruct a dot plot from raw dataConstruct an ordered stem plot from raw dataChoose an appropriate display for a variableFind the mean and the median, and choose between themFind measures of centre from a frequency displayWork backwards from a mean or medianFind the range, the quartiles and the IQRCalculate the standard deviation from the formulaApply the 68–95–99.7% ruleProduce a five-number summary and construct a boxplotFind the fences and identify outliersRead values and percentages from a boxplotCompare a numerical variable across groups