1. Lesson 2: Using the Normal Curve
Common Core Georgia Performance Standard
MCC9–12.S.ID.4★
Essential Questions
How is discrete data different from continuous data?
How can you tell if a set of values is normally distributed?
How can the standard normal distribution be used with a normal distribution that has a different mean and standard deviation?
Why is it a mistake to use the standard normal distribution to make decisions about data that are not normally distributed?
1.2.1: Normal Distributions and the 68–95–99.7 Rule

2. Vocabulary
rule, continuous data, continuous distribution, discrete data, Empirical Rule, interval, mean, median, mu, negatively skewed
1.2.1: Normal Distributions and the 68–95–99.7 Rule

3. Vocabulary
normal distribution, outlier, population, positively skewed, probability distribution, random variable, sample, sigma (lower and uppercase), skewed (right & left)
1.2.1: Normal Distributions and the 68–95–99.7 Rule

4. Vocabulary
standard deviation, standard normal distribution, summation notation, symmetric distribution, uniform distribution and z-score
1.2.1: Normal Distributions and the 68–95–99.7 Rule

5. Introduction
Probability distributions are useful in making decisions in many areas of life, including business and scientific research. The normal distribution is one of many types of probability distributions, and perhaps the one most widely used. Learning how to use the properties of normal distributions will be a valuable asset in many careers and subjects, including economics, education, finance, medicine, psychology, and sports.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

6. Introduction, continued
Understanding a data set requires finding four key components:
the overall shape of the distribution
a measure of central tendency or average
a measure of variation
a measure of population or sample size
The first three components are used in determining proportions and probabilities associated with values in normal distributions. The two main classes of data are discrete and continuous. We will begin by focusing on continuous distributions, particularly the normal distribution.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

7. Key Concepts
Understanding a data set, and how an individual value relates to the data set, requires information about the overall shape of the distribution as well as measures of center, measures of variation, and population (or sample) size. There are two types of data: discrete and continuous.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

8. Key Concepts, continued
Discrete data refers to a set of values with gaps between successive values.
For example, if you hire a bus with 65 seats for a field trip, but 82 people sign up to go on the field trip, you need more seats. You would increase the number of buses from one bus to two buses, rather than from one bus to a fraction of a second bus.
When using discrete data, we can assign probabilities to individual values. For example, the probability of rolling a 6 on a fair die is .
1.2.1: Normal Distributions and the 68–95–99.7 Rule

9. Key Concepts, continued
In contrast, continuous data is a set of values for which there is at least one value between any two given values—there are no gaps. For example, if a car accelerates from 30 miles per hour to 40 miles per hour, the car passes through every speed between 30 and 40 miles per hour. It does not skip instantly from 30 miles per hour to 40.
When using continuous data, we need to assign probabilities to an interval or range of values.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

10. Key Concepts, continued
For continuous data, the probability of an exact value is essentially 0, so we must assign a range or an interval of interest to calculate probability. For example, a car will accelerate through a series of speeds in miles per hour, including an infinite number of decimals. Because there are an infinite number of values between the starting speed and the desired speed, the probability of determining an exact speed is essentially 0.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

11. Key Concepts, continued
An interval is a range or a set of values that starts with a specified value, ends with a specified value, and includes every value in between. The starting and ending values are the limits, or boundaries, of the interval.
In other words, an interval is a set of values between a lower bound and an upper bound. The size of the interval depends on the situation being observed.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

12. Key Concepts, continued
The probability that a randomly selected student from a given high school is exactly 64 inches tall is effectively 0, since methods of measuring are not completely precise. Measuring tapes and rulers can vary slightly, and when we take measurements, we often round to the nearest quarter inch or eighth of an inch; it is impossible to determine a person’s height to the exact decimal place. However, we can determine the probability that a student’s height falls between two values, such as 63.5 and 64.5 inches, since this interval includes all of the infinite decimal values between these two heights.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

13. Key Concepts, continued
To determine the probability of an outcome using continuous data, we use the proportion of the area under the normal curve associated with the distribution of that data.
A normal curve is a symmetrical curve representing the normal distribution.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

14. Key Concepts, continued
A probability distribution is a graph of the values of a random variable with associated probabilities.
A random variable is a variable with a numerical value that changes depending on each outcome in a sample space. A random variable can take on different values, and the value that a random variable takes is associated with chance.
The area under a probability distribution is equal to 1; that is, 100% of all possible data values within the interval are represented under the curve.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

15. Key Concepts, continued
A continuous distribution is a graphed set of values (a curve) in a continuous data set.
We will examine two types of continuous distributions: uniform and normal.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

16. Key Concepts, continued
Continuous Uniform Distributions
A uniform distribution is a set of values that are continuous, are symmetric to a mean, and have equal frequencies corresponding to any two equally sized intervals.
In other words, the values are spread out uniformly throughout the distribution.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

17. Key Concepts, continued
To determine the probability of an outcome using a uniform distribution, we calculate the ratio of the width of the interval of interest for the given outcome to the overall width of the distribution:
The result of this proportion is equal to the probability of the outcome.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

18. Key Concepts, continued
In the uniform distribution below, the data values are spread evenly from 1 to 9:
1.2.1: Normal Distributions and the 68–95–99.7 Rule

19. Key Concepts, continued
Continuous Normal Distributions
Another type of a continuous distribution is a normal distribution.
A normal distribution is a set of values that are continuous, are symmetric to the mean, and have higher frequencies in intervals close to the mean than equal-sized intervals away from the mean. When graphed, data following a normal distribution forms a normal curve.
Normal distributions are symmetric to the mean. This means that 50% of the data is to the right of the mean and 50% of the data is to the left of the mean.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

20. Key Concepts, continued
The mean is a measure of center in a set of numerical data, computed by adding the values in a data set and then dividing the sum by the number of values in the data set.
The population mean is denoted by the Greek lowercase letter mu, μ, whereas the sample mean is denoted by
1.2.1: Normal Distributions and the 68–95–99.7 Rule

21. The standard deviation is a measure of average variation about a mean.
Key Concepts, continued
A population is made up of all of the people, objects, or phenomena of interest in an investigation. A sample is a subset of the population—that is, a smaller portion that represents the whole population.
The standard deviation is a measure of average variation about a mean.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

22. Key Concepts, continued
Technically, the standard deviation is the square root of the average squared difference from the mean, and is denoted by the lowercase Greek letter sigma, .
Steps to Find the Standard Deviation
Calculate the difference between the mean and each number in the data set.
Square each difference.
Find the mean of the squared differences.
Take the square root of the resulting number.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

23. Key Concepts, continued
Approximately 68% of the values in a normal distribution are within one standard deviation of the mean. Written as an equation, this is In other words, the mean, μ, plus or minus the standard deviation σ times 1 is approximately equal to 68% of the values in the distribution.
In the graph that follows, the shading represents these 68% of values that fall within one standard deviation of the mean.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

24. Key Concepts, continued Data Within One Standard Deviation of the Mean
1.2.1: Normal Distributions and the 68–95–99.7 Rule

25. Key Concepts, continued
Approximately 95% of the values in a normal distribution are within two standard deviations of the mean, as shown by the shading in the graph on the next slide.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

26. Data Within Two Standard Deviations of the Mean
Key Concepts, continued
Data Within Two Standard Deviations of the Mean
1.2.1: Normal Distributions and the 68–95–99.7 Rule

27. Key Concepts, continued
Approximately 99.7% of the values in a normal distribution are within three standard deviations of the mean, as shaded in the graph on the next slide.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

28. Data Within Three Standard Deviations of the Mean
Key Concepts, continued
Data Within Three Standard Deviations of the Mean
1.2.1: Normal Distributions and the 68–95–99.7 Rule

29. These percentages of data under the normal curve
Key Concepts, continued
These percentages of data under the normal curve
follow what is called the 68–95–99.7 rule. This rule is also known as the Empirical Rule.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

30. Key Concepts, continued
The standard normal distribution has a mean of 0 and a standard deviation of 1. A normal curve is often referred to as a bell curve, since its shape resembles the shape of a bell. Normal distribution curves are a common tool for teachers who want to analyze how their students performed on a test. If a test is “fair,” you can expect a handful of students to do very well or very poorly, with most scores being near average—a normal curve. If the curve is shifted strongly toward the lower or higher ends of the scores, then the test was too hard or too easy.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

31. Common Errors/Misconceptions
applying the 68–95–99.7 rule to distributions that are not normally distributed
assuming that all normal distributions have a mean of 0 and/or a standard deviation of 1
not applying symmetry in a normal distribution to calculate probabilities
1.2.1: Normal Distributions and the 68–95–99.7 Rule

32. Guided Practice
Example 2
Madison needs to ride a shuttle bus to reach an airport terminal. Shuttle buses arrive every 15 minutes, and the arrival times for buses are uniformly distributed. What is the probability that Madison will need to wait more than 6 minutes for the bus?
1.2.1: Normal Distributions and the 68–95–99.7 Rule

33. Guided Practice: Example 2, continued
Sketch a uniform distribution and shade the area of the interval of interest.
Start by drawing a number line.
The interval of the distribution goes from 0 minutes to 15 minutes, and the interval of interest is from 6 to 15. Shade the region between 6 and 15.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

34. Guided Practice: Example 2, continued
1.2.1: Normal Distributions and the 68–95–99.7 Rule

35. Guided Practice: Example 2, continued
Determine the total width of the distribution.
We can see that the total width of the distribution is 15 minutes.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

36. Guided Practice: Example 2, continued
Determine the width of the interval of interest.
Find the absolute value of the difference of the endpoints of the interval of interest.
The width of the interval of interest is 9 minutes.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

37. Guided Practice: Example 2, continued
Determine the proportion of the area of the interval of interest to the total area of the distribution.
Create a ratio comparing the area that corresponds to arrival times between 6 and 15 minutes to the area of the total time frame of 15 minutes between buses.
The proportion of the area of interest to the total area of the distribution is equal to the area of interest divided by the total area of the distribution.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

38. Guided Practice: Example 2, continued
The proportion of the area of interest to the total area of the distribution is 0.6.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

39. ✔ Guided Practice: Example 2, continued
Interpret the proportion in terms of the context of the problem.
The probability that Madison will wait more than 6 minutes for the bus is 0.6. ✔
1.2.1: Normal Distributions and the 68–95–99.7 Rule

40. Guided Practice: Example 2, continued
1.2.1: Normal Distributions and the 68–95–99.7 Rule

41. Guided Practice
Example 4
The scores of a particular college admission test are normally distributed, with a mean score of 30 and a standard deviation of 2. Erin scored a 34 on her test. If possible, determine the percent of test-takers whom Erin outperformed on the test.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

42. Guided Practice: Example 4, continued
Sketch a normal curve and shade the area of the interval of interest.
To sketch the normal curve, start by drawing a number line with a range of values that includes Erin’s score, 34. The mean of the scores is 30, so the middle number on the line will be 30. A range of 24 to 36 will give us a number line that has the mean, 30, in the middle, and an even number of data points on either side of the mean.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

43. Guided Practice: Example 4, continued
We want to know how many test-takers had scores lower than Erin’s. Erin scored a 34; therefore, the area of interest is the area to the left of 34.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

44. Guided Practice: Example 4, continued
Determine how many standard deviations away from the mean Erin’s score is.
From the problem statement, we know that Erin scored a 34, the mean is 30, and the standard deviation is 2. Erin’s score is greater than the mean.
Also, we can determine that Erin scored two standard deviations above the mean.
μ = (2) = 32
μ + 2σ = (2) = 34 Erin’s score
μ = (3) = 36
1.2.1: Normal Distributions and the 68–95–99.7 Rule

45. Guided Practice: Example 4, continued
Use symmetry and the 68–95–99.7 rule to determine the area of interest.
We know that the data in a normal curve is symmetrical about the mean. Since the area under the curve is equal to 1, the area to the left of the mean is 0.5, as shaded in the graph on the next slide.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

46. Guided Practice: Example 4, continued
Erin’s score is above the mean; therefore, we need to determine the area between the mean and Erin’s score and add it to the area below the mean to find the total area of interest.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

47. Guided Practice: Example 4, continued
Recall that the 68–95–99.7 rule states the percentages of data under the normal curve are as follows:
We know that We have already accounted for the area to the left of the mean, which includes from the mean down to –2σ. Since we found that Erin’s score is two standard deviations from the mean, we need to determine the area from the mean up to +2σ.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

48. Guided Practice: Example 4, continued
Since data is symmetric about the mean, we know that half of the area encompassed between μ ± 2 is above the mean. Therefore, divide 0.95 by 2.
The following graph shows the shaded area of interest to the right of the mean up until Erin’s score of 34.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

49. Guided Practice: Example 4, continued
Add the two areas together to get the total area below 2σ, which is equal to Erin’s score of 34.
= 0.975
The total area of interest for this data is
1.2.1: Normal Distributions and the 68–95–99.7 Rule

50. Guided Practice: Example 4, continued
A graphing calculator can also be used to calculate the area of interest.
On a TI-83/84:
Step 1: Press [2ND][VARS] to bring up the distribution menu.
Step 2: Arrow down to 2: normalcdf. Press [ENTER].
Step 3: Enter the following values for the lower bound, upper bound, mean (μ), and standard deviation (σ). Press [ENTER] after typing each value to navigate between fields. Lower: [(–)][99]; upper: [34]; μ: [30]; σ: [2].
Step 4: Press [ENTER] twice to calculate the area of interest.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

51. Guided Practice: Example 4, continued
On a TI-Nspire:
Step 1: Press the [home] key.
Step 2: Arrow over to the spreadsheet icon and press [enter].
Step 3: Press the [menu] key. Arrow down to 4: Statistics, then arrow right to bring up the sub-menu. Arrow down to 2: Distributions and press [enter].
Step 4: Arrow down to 2: Normal Cdf. Press [enter].
1.2.1: Normal Distributions and the 68–95–99.7 Rule

52. Guided Practice: Example 4, continued
Step 5: Enter the values for the lower bound, upper bound, mean (μ), and standard deviation (σ), using the [tab] key to navigate between fields. Lower Bound: [(–)][99]; Upper Bound: [34]; μ: [30]; σ: [2]. Tab down to “OK” and press [enter].
Step 6: The values entered will appear in the spreadsheet. Press [enter] again to calculate the probability.
The result from the graphing calculator verifies the area of interest is
1.2.1: Normal Distributions and the 68–95–99.7 Rule

53. ✔ Guided Practice: Example 4, continued
Interpret the proportion in terms of the context of the problem.
Convert the area of interest to a percent.
0.975 = 97.5%
Erin outperformed 97.5% of the students who also took the exam. ✔
1.2.1: Normal Distributions and the 68–95–99.7 Rule

54. Guided Practice: Example 4, continued
1.2.1: Normal Distributions and the 68–95–99.7 Rule