In the Dominican Republic in August, the distribution of daily high temperature is approximately normal with mean 86 degrees Fahrenheit (°F). Approximately 95% of all daily high temperatures are between 83°F and 89°F. What is the standard deviation of the distribution?

The standard deviation of the distribution calculated using the empirical relation is 1.5°F

According to the empirical rule :

95% of the distribution falls within 2 standard deviations from the mean

This can be defined as :

Mean ± 2(standard deviation)

Mean temperature = 86°

95% of temperature = 83°F to 89°F

We could use either the lower or upper boundary value to for our calculation :

Using the upper boundary value = 89°F

89° = mean + 2(standard deviation)

89° = 86 + 2σ

2σ = 89 - 86

2σ = 3

σ = 3 / 2

σ = 1.5°F

Therefore, the standard deviation of the distribution is 1.5°F

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The standard deviation of the distribution is 1.5ºF.

Explanation:

The known 68-95-99.7 rule, the empirical rule, states that, in a normal distribution 68% of the data are within one standard deviation from the mean, 95% of the data are within two standard deviations from the mean, and 99.7% are within three standard deviations:

68% ⇒ mean ± 1 standard deviation95% ⇒ mean ± 2 standard deviations99.7% ⇒ mean ± 3 standard deviation.

Hence, for the approximately normal distribution of the daily high temperatures, with mean 86º, and a range of 83ºF to 89ºF for 95% of the data, you can write:

86ºF ± 2 SD ⇒ 86ºF - 2 SD = 83ºF, and 86ºF ± 2 SD ⇒  86ºF + 2SD = 89 ºF

Both equations will lead to the same result:

86ºF - 2 SD  = 83ºF ⇒ 2 SD = 86ºF - 83ºF = 3ºF

         SD = 3ºF / 2 = 1.5ºF

Also:

86ºF + 2SD = 89ºF ⇒ 2 SD = 89ºF - 86ºF = 3ºF

         SD = 3ºF / 2 = 1.5ºF

Therefore, the standard deviation of the distribution is 1.5ºF.

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