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6. Descriptive Statistics
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Integrated Math 3

6.07 Normal distributions

Lesson
Practice
Lesson

Concept summary

When we have a large set of data, it often happens that most of the measurements will be clustered close to the mean value with the density of the observations lessening with we move away from the mean in either direction. Results of this kind displayed in a histogram show a central peak with columns of decreasing height on each side of the mean.

A data set that is symmetrical and bell-shaped about the mean, and which meets certain more precise shape requirements, is said to have an approximately normal distribution.

The shape of the normal distribution will depend on the population parameters: mean \mu and standard deviation \sigma.

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A small standard deviation provides a tight cluster around the mean
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A larger standard deviation shows data that is more spread out
Empirical Rule {(68-95-99.7\%)}

A statistical rule that provides an estimate for the distribution of approximately normal data.

In a density curve, the area beneath the curve represents probability with the area under the entire curve equal to 100\%, or 1. When data is approxiately normally distributed, the probability between 1, 2, and 3 standard deviations can be accurately summarized using the empirical rule.

Worked examples

Example 1

The grades on a recent exam are approximately normally distributed with a mean score of 72 and a standard deviation of 4.

a

Construct a normal curve and label the boundaries for the empirical rule.

Approach

A normal curve will have a symmetric bell-like appearance with the mean as the central value and show markings for \text{mean} \pm 1 \text{sd}, \text{mean} \pm 2 \text{sd}, and \text{mean} \pm 3 \text{sd}.

Solution

b

Find the percentage of students who scored between 64 and 68 on the exam.

Approach

To use the empirical rule, we must first determine how many standard deviations 64 and 68 are away from the mean score of 72.

64 is two standard deviations below the mean and 68 is one standard deviation below the mean.

Solution

The probability of a value between 1 and 2 standard deviations below the mean is 13.5\%.

Reflection

We can shade the bell curve to model this solution and as a way to check the reasonableness of the value.

c

If 32 students took the exam, determine the number of students expected to score 80 or more on the exam.

Approach

We first need to determine the number of standard deviations 80 is away from the mean score of 72. Then, we can multiply the probability by 32 students to determine the number of students who may have scored more than 80.

Solution

80 is two standard deviations above the mean. According to the empirical rule, the probability of being more than 2 standard deviations above the mean is \dfrac{1-.95}{2}=.025 or 2.5\%.

There are 32(.025)=0.8. If the data is approximately normal, not even one student will score above an 80 in a class of 32.

Reflection

Use the normal curve to check the reasonableness of this solution:

Outcomes

M3.N.Q.A.1

Use units as a way to understand real-world problems.*

M3.N.Q.A.1.D

Choose an appropriate level of accuracy when reporting quantities.

M3.S.ID.A.4

Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages using the Empirical Rule.*

M3.MP1

Make sense of problems and persevere in solving them.

M3.MP2

Reason abstractly and quantitatively.

M3.MP3

Construct viable arguments and critique the reasoning of others.

M3.MP4

Model with mathematics.

M3.MP5

Use appropriate tools strategically.

M3.MP6

Attend to precision.

M3.MP7

Look for and make use of structure.

M3.MP8

Look for and express regularity in repeated reasoning.

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