Download Chapter 5C: Normal Curves. Standard Deviations and Percentiles of

Chapter 5C: Normal Curves.
Standard Deviations and Percentiles of a Symmetric Distribution The standard deviation of a histogram
measures how your data values deviate from the mean (which is sometimes denoted as µ or x). Standard deviations, along with the mean describe variability your observed values, much like with the quartiles and the median.
The standard deviation (usually denoted as σ) can be computed from a formula (see p. 184). You will not have
to compute this value, but you will need to be able to use and interpret it, as illustrated below. The following
table lists the heights of the students in one 5th grade classroom:
Example 1: Heights of 37 5th Grade Students.
40”: 1
45”: 2
46”: 3
47”: 4
48”: 6
49”: 7
50”: 6
51”: 4
52”: 3
57”: 1
Answer the following:
1. Compute the mean of the data.
2. The standard deviation of this data is σ = 2.75. How many individuals are within one standard deviation
of your mean? How many individuals are within two standard deviations of your mean? How many are three
standard deviations from your mean?
An outlier is a data point in your histogram that lies beyond three standard deviations of your mean. Do we have
any such outliers in our data?
Percentiles: You often hear the word percentile thrown around in statistics. In a distribution, the percentile
of one data entry is the percentage of the entries that are less than or equal to its value. So if your score of
80 on an exam puts you in the 75th percentile, that means that you did as well or better as 75% of your classmates.
Example 2:
Going back to our 5th grade students, determine the percentile of each student listed below:
a. Alan, height: 46”
b. Britney, height: 45”
c. Charlie, height: 49”
d. Dorothy, height: 51”
e. Ethan, height: 57”
Bell Shaped Curves and Normal Distributions
Many histograms are symmetric, single-peaked, and has a distinct bell shape. Statisticians usually super impose
a smooth curve over these histograms. The curve is an idealized description of the distribution. Such a curve
is called a normal curve and a distribution whose shape is described by that curve is called a normal distribution.
Characteristics of a normal curve:
1.
2.
3.
4.
5.
The x-value corresponding to the highest point in the middle is the mean.
The area of the curve below the x-axis sums up to 1.00, or 100
68% of the total area of the curve will fall one standard deviation away from the mean.
95% of the total area of the curve will fall two standard deviations away from the mean.
99.7% of the total area of the curve will fall three standard deviations away from the mean.
Collectively, the last three items listed above is called the Empirical Formula, or the 68 - 95 - 99.7 rule.
Example 3: Analyzing the Normal Curve and the Empirical Formula.
Compute the areas of each of the shaded regions. Based on this result, in a normal distribution, what percent of
our data should we expect to be outliers?
Example 4: Grading on a Curve. Some teachers grade on a bell curve”. One way to do this is to assign a
”C” to students that fall one standard deviation away from the mean. Students that scored between one and
two standard deviations above the mean would receive a ”B”, and students who scored more than two standard
deviations above the mean would receive an ”A”. The teacher would then symmetrically assign ”D”s and ”F”s.
Suppose that there were 200 students in the class and that the scores followed the Empirical formula. Determine
the class grade breakdown.
Example 5: Percentiles and Normal Distributions:
Compute the percentile of the following values, where µ is the mean and σ = standard deviation:
a. µ − 3σ
b. µ − 2σ
c. µ − 1σ
d. µ + 1σ
e. µ + 2σ
f. µ + 3σ