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Empirical Rule Calculator

Find the share of a normal distribution within one, two and three standard deviations of the mean, and place a value on the curve.

What this tool does

Find the share of a normal distribution within one, two and three standard deviations of the mean, and place a value on the curve. You fill in these fields: Mean, Standard deviation, and Value to place (optional). Some fields are optional. It shows within one standard deviation, Z-score, and percent at or below, plus a table (bands around the mean).

Inputs

  • Mean: A number. Starts at 100. The average of the data, in the same unit as the standard deviation.
  • Standard deviation: A number. Must be greater than 0. Starts at 15. How spread out the data is, in the same unit as the mean.
  • Value to place (optional): A number. Optional. A value from the same data, to see how many standard deviations it sits from the mean.

How to use

  1. Enter the mean.
  2. Enter the standard deviation.
  3. Enter the value to place (optional).
  4. Read the result: within one standard deviation, Z-score, and percent at or below, plus a table (bands around the mean). It appears once every required field has a valid value.

Worked examples

Example 1. Inputs:

  • Mean: 100
  • Standard deviation: 15

Results:

  • Within one standard deviation: 68.27 % of values
  • Bands around the mean, row 1 of 3: Standard deviations either side 1; From 85; To 115; Rule of thumb (%) 68; Exact (%) 68.27
  • Bands around the mean, row 2 of 3: Standard deviations either side 2; From 70; To 130; Rule of thumb (%) 95; Exact (%) 95.45
  • Bands around the mean, row 3 of 3: Standard deviations either side 3; From 55; To 145; Rule of thumb (%) 99.7; Exact (%) 99.73

How this is worked out: Bands are mean plus or minus k x sd. k = 1: 100 - 15 = 85 and 100 + 15 = 115. k = 2: 70 and 130. k = 3: 55 and 145. Exact share within k sd is erf(k / sqrt 2): erf(0.707107) = 0.682689, erf(1.414214) = 0.954500, erf(2.121320) = 0.997300, so 68.2689, 95.4500 and 99.7300 percent.

Example 2. Inputs:

  • Mean: 50
  • Standard deviation: 10
  • Value to place (optional): 65

Results:

  • Within one standard deviation: 68.27 % of values
  • Bands around the mean, row 1 of 3: Standard deviations either side 1; From 40; To 60; Rule of thumb (%) 68; Exact (%) 68.27
  • Bands around the mean, row 2 of 3: Standard deviations either side 2; From 30; To 70; Rule of thumb (%) 95; Exact (%) 95.45
  • Bands around the mean, row 3 of 3: Standard deviations either side 3; From 20; To 80; Rule of thumb (%) 99.7; Exact (%) 99.73
  • Z-score: 1.50 (Standard deviations from the mean)
  • Percent at or below: 93.32% (Share of the distribution at or below this value)
  • Note: Within 2 standard deviations of the mean.

How this is worked out: z = (65 - 50) / 10 = 1.5. The percent at or below is the normal cumulative value Phi(1.5) = 0.9331928, so 93.31928 percent. 1.5 lies above 1 and at or below 2, so band 2, and it is above the mean.

Example 3. Inputs:

  • Mean: 0
  • Standard deviation: 1
  • Value to place (optional): -3.5

Results:

  • Within one standard deviation: 68.27 % of values
  • Bands around the mean, row 1 of 3: Standard deviations either side 1; From -1; To 1; Rule of thumb (%) 68; Exact (%) 68.27
  • Bands around the mean, row 2 of 3: Standard deviations either side 2; From -2; To 2; Rule of thumb (%) 95; Exact (%) 95.45
  • Bands around the mean, row 3 of 3: Standard deviations either side 3; From -3; To 3; Rule of thumb (%) 99.7; Exact (%) 99.73
  • Z-score: -3.50 (Standard deviations from the mean)
  • Percent at or below: 0.02% (Share of the distribution at or below this value)
  • Note: More than 3 standard deviations below the mean.

How this is worked out: z = (-3.5 - 0) / 1 = -3.5. Phi(-3.5) = 0.000232629, so 0.0232629 percent. 3.5 is more than 3, so band 4, below the mean.

Example 4. Inputs:

  • Mean: 0
  • Standard deviation: 2
  • Value to place (optional): 0

Results:

  • Within one standard deviation: 68.27 % of values
  • Bands around the mean, row 1 of 3: Standard deviations either side 1; From -2; To 2; Rule of thumb (%) 68; Exact (%) 68.27
  • Bands around the mean, row 2 of 3: Standard deviations either side 2; From -4; To 4; Rule of thumb (%) 95; Exact (%) 95.45
  • Bands around the mean, row 3 of 3: Standard deviations either side 3; From -6; To 6; Rule of thumb (%) 99.7; Exact (%) 99.73
  • Z-score: 0.00 (Standard deviations from the mean)
  • Percent at or below: 50.00% (Share of the distribution at or below this value)
  • Note: Within 1 standard deviation of the mean.

How this is worked out: A value equal to the mean has z = 0 / 2 = 0. Half the distribution lies at or below the mean, so 50 percent. Band 1 is the narrowest band containing 0, and the side is at the mean. Band 1 runs from 0 - 2 = -2 to 0 + 2 = 2.

Limitations

  • Assumes the data follow a normal distribution. For skewed data the percentages will not match your data.
  • The percentages describe the ideal normal curve, not your data. Check the shape of your data before relying on them.
  • Needs the mean and standard deviation. It does not work them out from a list of values.

FAQs

Q: What does the empirical rule say?
A: For a normal, bell-shaped distribution, about 68 percent of values lie within one standard deviation of the mean, about 95 percent within two, and about 99.7 percent within three.
Q: Why do the exact percentages differ slightly from 68, 95 and 99.7?
A: The exact shares for a normal distribution are 68.27, 95.45 and 99.73 percent. The rule of thumb rounds them. Both columns are shown in the table.
Q: What is a z-score?
A: It is the number of standard deviations a value sits above or below the mean. It is the value minus the mean, divided by the standard deviation.
Q: What does percent at or below mean?
A: It is the share of a normal distribution that lies at or below the value you entered. It is worked out from the normal curve, not from your data.

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