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Suppose a normally distributed set of data has a mean of 108 and a standard deviation of 12.

Use the 68-95-99.7 Rule to determine the percent of scores in the data set expected to be below a score of 120.

Give your answer as a percent and include as many decimal places as the 68-95-99.7 rule dictates.

Caution: Using tables or Excel for this may produce a wrong answer. Use the 68-95-99.7 rule.

Answer :

By using the 68-95-99.7 Rule, the percent of scores in the data set that are expected to be below a score of 120 is 84.1%.

What is the 68-95-99.7 Rule?

In Mathematics, the 68-95-99.7 Rule is sometimes referred to as the Empirical Rule and it can be defined as a statistical rule which states that:

The middle 68% of a normal distribution would be within one (1) standard deviation of its mean i.e between μ - σ and μ + σ.

The middle 95% of a normal distribution would be within two (2) standard deviations of its mean i.e between μ - 2σ and μ + 2σ.

The middle 99.7% of a normal distribution would be within three (3) standard deviations of its mean i.e between μ - 3σ and μ + 3σ.

At the middle 68% of the normal distribution, we would apply the 68-95-99.7 Rule to the given data as follows;

μ - σ = 108 - 12 = 96.

μ - σ = 108 + 12 = 120.

Where:

  • μ is the sample mean.
  • x is the score.
  • σ is the standard deviation.

Now, we can determine the percent of scores (P) in the data set that are expected to be below a score (x) of 120 by using the probabilities and standard distribution table;

P(x < 120) = P((x - μ)/σ < (120 - 108)/12)

P(x < 120) = P((x - μ)/σ < 12/12)

P(x < 120) = P(z < 1)

P(x < 120) = 0.841

P(x < 120) = 84.1%

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