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Empirical Rule Calculator





Answer:

For a bell-shaped (normal) distribution:

Approximately 68% of the data values will fall within 1 standard deviation of the mean, from $72$ to $132$.

Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $42$ to $162$.

Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 12$ to $192$.


Solution:

The first part of the empirical rule states that 68% of the data values will fall within 1 standard deviation of the mean. To calculate "within 1 standard deviation," you need to subtract 1 standard deviation from the mean, then add 1 standard deviation to the mean. That will give you the range for 68% of the data values. $$ 102 - 30 = 72 $$ $$ 102 + 30 = 132 $$ The range of numbers is 72 to 132


The second part of the empirical rule states that 95% of the data values will fall within 2 standard deviations of the mean. To calculate "within 2 standard deviations," you need to subtract 2 standard deviations from the mean, then add 2 standard deviations to the mean. That will give you the range for 95% of the data values. $$ 102 - 2 \cdot 30 = 42 $$ $$ 102 + 2 \cdot 30 = 162 $$ The range of numbers is 42 to 162


Finally, the last part of the empirical rule states that 99.7% of the data values will fall within 3 standard deviations of the mean. To calculate "within 3 standard deviations," you need to subtract 3 standard deviations from the mean, then add 3 standard deviations to the mean. That will give you the range for 99.7% of the data values. $$ 102 - 3 \cdot 30 = 12 $$ $$ 102 + 3 \cdot 30 = 192 $$ The range of numbers is 12 to 192


Finally, we can use the symmetry of the bell curve to further divide up the percentages.

  • 2.35% of the data values will lie between 12 and 42
  • 13.5% of the data values will lie between 42 and 72
  • 34% of the data values will lie between 72 and 102
  • 34% of the data values will lie between 102 and 132
  • 13.5% of the data values will lie between 132 and 162
  • 2.35% of the data values will lie between 162 and 192
Have a look at the article below to understand where these percentages come from.

How to Use the Empirical Rule Calculator with Mean and Standard Deviation

You can use this Empirical Rule Calculator with mean and standard deviation to find the percent of data values between two numbers for bell-shaped distributions as well as a detailed solution.  First, enter your data set, and then you will get an explanation on the distribution of numbers based on the Empirical Rule.

Furthermore, you can use the Empirical Rule Calculator as a learning tool.  Begin by looking at a sample solution to a couple different problems and then work through the steps shown in the solution.  After you have a handle on the sample solutions, try a problem on your own using the same strategy, then check your work with the calculator.  You GOT this!

Use the empirical rule calculator to estimate percentages of values within 1, 2, and 3 standard deviations of the mean.

Afterward, you can take a look at Chebyshev’s Theorem Calculator. You can use that calculator for all types of distributions, so it’s ideal for unknown distribution shapes or skewed distributions.

What is the Empirical Rule?

We can use Empirical Rule in statistics, also known as the 68, 95, 99 rule, to estimate percentages between z-scores or between two raw scores.  With the Empirical Rule, we can estimate the percentages of data values up to 3 standard deviations away from the mean.  The Empirical Rule Calculator above will be able to tell you the percentage of values within 1, 2 or 3 standard deviations of the mean.

Within 1 standard deviation –  This refers to the range of values between a z-score of -1 to a z-score of +1.

Within 2 standard deviations – This refers to the range of values between a z-score of -2 to a z-score of +2.

And finally, within 3 standard deviations – This refers to the range of values between a z-score of -3 to a z-score of +3.

Since the symbol for the population standard deviation is “sigma,” $ \sigma $, we sometimes refer to the standard deviation as “sigma.”

  • z = -1 to z = +1 (within 1 sigma)
    Then,
  • z = -2 to z = +2 (within 2 sigma)
    Lastly,
  • z = -3 to z = +3 (within 3 sigma)

The Percentage Rules

The Empirical Rule is broken down into three percentages, 68, 95, and 99.7.  Hence, it’s sometimes called the 68 95 and 99.7 rule.  The first part of the rule states:

68% of the data values in a normal, bell-shaped, distribution will lie within 1 standard deviation (within 1 sigma) of the mean.

Next, the second part of the rule states:

95% of the data values in a normal, bell-shaped, distribution will lie within 2 standard deviation (within 2 sigma) of the mean.

Finally, the third part of the rule states:

99.7% of the data values in a normal, bell-shaped, distribution will lie within 3 standard deviation (within 3 sigma) of the mean.

The picture below is helpful for understanding the Empirical Rule, and so it’s a good idea to sketch this diagram whenever you need to complete an Empirical Rule problem.  The red area refers to the percentage of scores within 1 sigma.  The blue area refers to the percentage of scores within 2 sigma.  The black area refers to the percentage of scores within 3 sigma.

empirical rule with z-scores 68 95 and 99.7 percentages
Empirical Rule on a Bell Shaped Distribution with Z-Scores 

How to Calculate the Empirical Rule

To calculate the empirical rule, you need to be provided with a mean and standard deviation for a bell-shaped, normal distribution.  Otherwise, you can also use z-scores with the empirical rule.  In that case, the mean z-score is 0 and the standard deviation is 1.  However, most statistics problems involving the Empirical Rule will provide a mean and standard deviation.

Suppose you are provided with a bell-shaped, normal distribution that has a mean, $\mu$, of 50, and a standard deviation, $\sigma%$, of 5.  To apply the Empirical Rule, add and subtract up to 3 standard deviations from the mean.  This is exactly how the Empirical Rule Calculator finds the correct ranges.

FIRST PART:  First, subtract and add 1 standard deviation from/to the mean:

50 5 = 45

50 + 5 = 55

Therefore, 68% of the values fall between scores of 45 to 55.

SECOND PART:  Then, subtract and add 2 standard deviations from/to the mean:

50  (2)(5)

50  10 = 40

50 + (2)(5)

50 10 = 60

Therefore, 95% of the values fall between scores of 40 to 60.

THIRD PART:  Lastly, subtract and add 3 standard deviations from/to the mean:

50  (3)(5)

50  15 = 35

50 + (3)(5)

50 15 = 65

Therefore, 99.7% of the values fall between scores of 35 to 65.

Now, try inputing a mean of 50 and standard deviation of 5 into the Empirical Rule Calculator above to verify the ranges we just calculated by hand.

The picture below is helpful for understanding the Empirical Rule for this specific example.  The raw scores are labeled along the horizontal axis.

Empirical Rule with raw scores and percentages 68 95 and 99.7
Empirical Rule on a Bell-Shaped Distribution with Raw Scores

How to Use the Empirical Rule to Solve a Problem – Verified by the Empirical Rule Calculator

And now, we are going to look at a problem that requires the use of the Empirical Rule and demonstrate how to solve it.

Example:  Suppose a bell-shaped distribution of standardized test scores has a mean of 300 and a standard deviation of 22.  What percentage of the scores fall between 256 and 344?

First, calculate 1, 2, and 3 standard deviations below the mean, and 1, 2, and 3 standard deviations above the mean.

Calculations

1 standard deviation below the mean: 300 (1)(22) = 278
2 standard deviations below the mean: 300  (2)(22) = 256
3 standard deviations below the mean: 300  (3)(22) = 234

1 standard deviation above the mean: 300 + (1)(22) = 322
2 standard deviations above the mean: 300 + (2)(22) = 344
3 standard deviations above the mean: 300 + (3)(22) = 366

Draw a Picture

Next, draw a bell curve and label the center of the bell curve with the mean.  On the left side, label 1, 2, and 3 standard deviations below the mean with the values 278, 256, and 234.  On the right side, label 1, 2, and 3 standard deviations above the mean with the values 322, 344, and 366.

Empirical Rule with test scores 95 percent shaded
Label from 3 standard deviations below the mean to 3 standard deviations above the mean.

Some students get confused about labeling and mistakenly put the numbers on the horizontal axis out of order.  Make sure that your numbers are on the horizontal axis in increasing order from left to right.

Since we are asked for the percentage of scores between 256 and 344, shade the area under the bell curve between those values.  Now it’s clear based on the picture that we are asked for a percentage within 2 standard deviations of the mean (from 2 standard deviations below to 2 standard deviations above the mean).  And so, based on the Empirical Rule, that percentage is 95%.

The answer is: 95% of the test scores are between 256 and 344.

Now, use the Empirical Rule Calculator above to verify the range for 95% of the scores is 256 to 344.

Using the Symmetry of the Bell Curve to Further Divide Up Percentages

Using the symmetry of the bell curve, percentages can be broken down even further.  For example, the first part of the rule states that 68% of the data values are within 1 standard deviation.  That is, from z = -1 to z = +1.  Since the bell curve is symmetrical on the right and left, we can see that half of 68%, or 34%, of the data values lie from z = -1 to z = 0, and the other half, 34%, lie from z = 0 to z = +1.

Empirical Rule with center 68% divided into two halves 34% and 34%
Divide the center 68% into two halves, 34% and 34%.

 

Now let’s look at the second part of the rule.

This part states that 95% of the data values are within 2 standard deviations.  That is, from z = -2 to z = +2.  Since the bell curve is symmetrical on the right and left, we can see that half of 95%, or 47.5%, of the data values lie from z = -2 to z = 0, and the other half, 47.5%, lie from z = 0 to z = +2.  Subtracting 34% from 47.5%, we can see that 13.5% of the data values lie from z = -2 to z = -1, and 13.5% of the data values lie from z = +1 to z = +2.  Now, let’s expand our diagram with that information.

Empirical rule with divided 95% into 13.5% 34% 34% and 13.5%
Divide the center 95% into 13.5%, 34%, 34%, and 13.5%.

 

Lastly, let’s look at the third part of the rule.

This part states that 99.7% of the data values are within 3 standard deviations.  That is, from z = -3 to z = +3.  The percentage of area in the two remaining tails on the left and right must be 4.7%, since 99.7% – 95% is 4.7%.  Since the bell curve is symmetrical on the right and left, we can see that half of 4.7%, or 2.35%, of the data values lie from z = -3 to z = -2, and the other half, 2.35%, lie from z = 2 to z = 3.  Let’s expand our diagram with that information.

Empirical rule with divided 99.7% into 2.35% 13.5% 34% 34% 13.5% and 2.35%
Divide the center 99.7% into 2.35%, 13.5%, 34%, 34%, 13.5%, and 2.35%.

 

How to Use the Empirical Rule to Solve More Complex Problems

Let’s take the example we were working on before and apply this new information about the symmetry of the bell curve to answer the question.  The Empirical Rule Calculator can find the ranges for each of the six sections of the bell curve.  Then, it’s your job to sum up the corresponding percentages.  Afterward, see the What’s Next section below for information on the Z-Table and Normal CDF calculators for alternate ways to solve these types of problems.

Example:  Suppose a bell-shaped distribution of standardized test scores has a mean of 300 and a standard deviation of 22.  What percentage of the scores fall between:

  • 234 and 278?
  • 256 and 322?
  • 278 and 366?

Get started by first labeling 3 standard deviations below the mean to 3 standard deviations above the mean on a bell curve, just as in the example above.  Then, divide the bell curve into 6 parts and label the parts 2.35%, 13.5%, 34%, 34%, 13.5% and 2.35%.

Empirical rule with divided 99.7% into 2.35% 13.5% 34% 34% 13.5% and 2.35%
Label the horizontal axis with raw scores corresponding to z = -3 to z = +3. Divide the center 99.7% of area into 2.35%, 13.5%, 34%, 34%, 13.5%, and 2.35%.

 

234 and 278

A picture is so important here, so don’t skip drawing the diagram!  Looking at the diagram above, note all the areas labeled between 234 and 278.  They are 2.35% and 13.5%.  So to find the the answer to this problem, simply sum these two areas.

2.35% + 13.5% = 15.85%

Answer: 15.85% of the test scores are between 234 and 278.

256 and 322

Looking at the diagram above, note all the areas labeled between 256 and 322.  They are 13.5%, 34% and 34%.  Hence, the answer to this problem is the sum of these areas.

 13.5% + 34% + 34%= 81.5%

Answer: 81.5% of the test scores are between 256 and 322.

278 and 366

Looking at the diagram above, note all the areas labeled between 278 and 366.  They are 34%, 34%, 15.5%, and 2.35%.  Finally, get the answer to this problem by summing these areas.

 34% + 34% + 15.5% + 2.35% = 83.85%

Answer: 83.85% of the test scores are between 278 and 366.

Take the Empirical Rule Quiz

Empirical Rule Quiz

How well do you know the empirical rule? Take this quiz to find out!  

What’s Next

The Empirical Rule Calculator is a helpful tool for identifying the percentage of area under the curve in a bell-shaped, or normal, distribution.  However, Chebyshev’s Theorem is used for estimating area under the curve of a non bell-shaped distribution.  Therefore, you can use the Chebyshev’s Theorem Calculator for applying Chebyshev’s Rule. You’ll also get a step by step solution and practice examples as you’ve seen here for the Empirical Rule.

Because the Empirical rule gives percentages for 1, 2, and 3 standard deviations, the Empirical Rule Calculator is useful for finding percentages for whole standard deviations.  However, in statistics we often need to find area under the bell curve for standard deviations that are not whole, for example, from 2.35 standard deviations below the mean to 1.4 standard deviations above the mean.  So to find those areas, you can either use the Z-Table, or the Online Normal CDF calculator .

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