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LESSON 2.4 • The Empirical Rule and Assessing Normality 123

Let’s calculate the percent of data values within 1, 2, and 3 standard deviations of FYI
the mean:
The procedures to assess normality

Mean ±1SD: 106.883 1(19.484) 87.399 to 126.367 63 out of 77 = 81.8% described here are practical and
±

±

Mean ± 2SD: 106.883 2(19.484) 67.915 to 145.851 71 out of 77 = 92.2% accessible to students taking their

±

Mean ± 3SD: 106.883 3(19.484) 48.431 to 165.335 77 out of 77 = 100.0% first statistics course. There are more Lesson 2.4

In a normal distribution, about 68% of the values fall within 1 standard deviation advanced methods, including more

of the mean. For the cereal data, almost 82% of the brands had between 87.399 and than 10 different statistical tests, that
126.367 calories. These two percentages are far apart. So, this distribution of calories aren’t appropriate here. Two of the most
in breakfast cereals is not approximately normal.
common tests are the Shapiro–Wilk test
and the Anderson–Darling test.
EXAMPLE
(C) 2021 BFW Publishers -- for review purposes only.
AL TERNA TE EX AMPLE
Is the amount of cream in Oreo cookies normally distributed? What score for the floor?
Assessing normality Assessing normality

PROBLEM: Statistics students Ryan Wells and Allie Quiroz collected data on the PROBLEM: The final scores of the 36

weight of cream (in grams) in a random sample of 45 Double Stuf Oreo cookies as
part of their final project. Here are their data: Dan Anderson floor routines during the NCAA Women’s
6.37 6.44 6.44 6.45 6.48 6.49 6.49 6.52 6.58 16 Gymnastics Championship were
recorded for a recent year. A dotplot
6.59 6.62 6.62 6.63 6.67 6.67 6.68 6.69 6.71
14 and summary statistics for the data are
6.72 6.73 6.73 6.73 6.73 6.74 6.74 6.76 6.80 12
6.80 6.83 6.85 6.85 6.85 6.86 6.87 6.88 6.90 10 shown here. Is this distribution of scores
6.91 6.92 6.92 6.92 6.94 7.01 7.04 7.09 7.15 8 approximately normal? Justify your
A histogram and summary statistics for the Frequency 6 answer based on the graph and the
data are shown here. Is this distribution empirical rule.
of amount of cream in Double Stuf Oreo 4
cookies approximately normal? Justify 2
your answer based on the graph and the
empirical rule. 0
6.30 6.45 6.60 6.75 6.90 7.05 7.20 9.30 9.40 9.50 9.60 9.70 9.80 9.90
SOLUTION: Weight of cream (g) Floor routine score
The histogram looks roughly symmetric, single- n Mean SD Min Q 1 Med Q 3 Max

peaked, and somewhat bell-shaped. The percents 45 6.742 0.184 6.37 6.62 6.73 6.875 7.15
of values within 1, 2, and 3 standard deviations of n mean SD min Q1 med Q3 max
the mean are:
36 9.86 0.11 9.31 9.84 9.88 9.93 9.96
±

Mean1SD:6.742 ±1(0.184) 6.558 to 6.926 32 out of 45 71.1%= Never say that a distribution
±
=

Mean2SD:6.742 ±2(0.184) 6.374 to 7.110 43 out of 45 95.6% of quantitative data is normal. SOLUTION:

±

Mean3SD:6.742 ±3(0.184) 6.190 to 7.294 45 out of 45 100.0%= Real-world data always show at

least slight departures from a The dotplot is skewed left with a

These percents are fairly close to the 68% , 95% , and 99.7% normal distribution. The most you noticeable low outlier. The percents

targets for a normal distribution. The graphical and numerical can say is that a distribution is
evidence suggests that this distribution of amount of cream “approximately normal.” of values within 1, 2, and 3 standard
in Oreo Double Stuf cookies is approximately normal. deviations of the mean are:
FOR PRACTICE TRY EXERCISE 17.
±
Mean ±1SD: 9.86 1(0.11)
9.75to9.97 32out of 36 = 89%
±
Mean ± 2SD: 9.86 2(0.11)
9.64to10.08 35 outof36 = 97.2%
±
Mean ± 3SD: 9.86 3(0.11)
9.53to10.19 35 outof36 = 97.2%
03_StarnesSPA4e_24432_ch02_088_153.indd 123 07/09/20 1:56 PM
These percents are not very close to
the 68%, 95%, and 99.7% targets for a
normal distribution. (In particular, 89% is
not very close to 68%.) The graphical and
numerical evidence suggests that this
distribution of floor routine scores is not
approximately normal.
LESSON 2.4 • The Empirical Rule and Assessing Normality 123
03_TysonTEspa4e_25177_ch02_088_153_4pp.indd 123 10/11/20 7:46 PM