Are there any outliers?

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20, 22, 23, 24, 24, 25, 25, 27, 35

• Are there any outliers?
• Draw a skeleton boxplot.
• Draw a modified boxplot.

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Chebyshevs The Empirical Rule
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Describing Data in terms of the Standard
Deviation.
Test Mean 80 St. Dev. 5
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Chebyshevs Rule

• The percent of observations that are within k
  standard deviations of the mean is at least

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Facts about Chebyshev

• Applicable to any data set whether it is
  symmetric or skewed.
• Many times there are more than 75 - this is a
  very conservative estimation.

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St. Dev. w/in k st. dev. of
mean 2 3 4 4.472
5 10
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Interpret using Chebyshev
Test Mean 80 St. Dev. 5

1. What percent are between 75 and 85?
2. What percent are between 60 and 100?

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Collect wrist measurements (in)

• Create distribution
• Find st. dev mean.
• What percent is within 1 deviation of mean

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Practice Problems

• Using Chebyshev, solve the following problem for
  a distribution with a mean of 80 and a st. dev.
  Of 10.
• a. At least what percentage of values will fall
  between 60 and 100?
• b. At least what percentage of values will fall
  between 65 and 95?

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Normal Distributions

• These are special density curves.
• They have the same overall shape
• Symmetric
• Single-Peaked
• Bell-Shaped
• They are completely described by giving its mean
  (?) and its standard deviation (?).
• We abbreviate it N(?,?)

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Normal Curves.

• Changing the mean without changing the standard
  deviation simply moves the curve horizontally.
• The Standard deviation controls the spread of a
  Normal Curve.

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Standard Deviation

• Its the natural measure of spread for Normal
  distributions.
• It can be located by eye on a Normal curve.
• Its the point at which the curve changes from
  concave down to concave up.

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Why is the Normal Curve Important?

• They are good descriptions for some real data
  such as
• Test scores like SAT, IQ
• Repeated careful measurements of the same
  quantity
• Characteristics of biological populations
  (height)
• They are good approximations to the results of
  many kinds of chance outcomes
• They are used in many statistical inference
  procedures.

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Empirical Rule

• Can only be used if the data can be reasonably
  described by a normal curve.
• Approximately
• 68 of the data is within 1 st. dev. of mean
• 95 of the data is within 2 st. dev. of mean
• 99.7 of data is within 3 st. dev. of mean

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Empirical Rule

• What percent do you think
• www.whfreeman.com/tps4e

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Empirical Rule (68-95-99.7 Rule)

• In the Normal distribution with mean (?) and
  standard deviation (?)
• Within 1? of ? 68 of the observations
• Within 2? of ? 95 of the observations
• Within 3? of ? 99.7 of the observations

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The distribution of batting average (proportion
of hits) for the 432 Major League Baseball
players with at least 100 plate appearances in
the 2009 season is normally distributed defined
N(0.261, 0.034).

• Sketch a Normal density curve for this
  distribution of batting averages. Label the
  points that are 1, 2, and 3 standard deviations
  from the mean.
• What percent of the batting averages are above
  0.329?
• What percent are between 0.227 and .295?

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Scores on the Wechsler adult Intelligence Scale
(a standard IQ test) for the 20 to 34 age group
are approximately Normally distributed. N(110,
25).

• What percent are between 85 and 135?
• What percent are below 185?
• What percent are below 60?

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• A sample of the hourly wages of employees who
  work in restaurants in a large city has a mean of
  5.02 and a st. dev. of 0.09.
• a. Using Chebyshevs, find the range in which
  at least 75 of the data will fall.
• b. Using the Empirical rule, find the range in
  which at least 68 of the data will fall.

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The mean of a distribution is 50 and the standard
deviation is 6. Using the empirical rule, find
the percentage that will fall between 38 and 62.
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A sample of the labor costs per hour to assemble
a certain product has a mean of 2.60 and a
standard deviation of 0.15, using Chebyshevs,
find the values in which at least 88.89 of the
data will lie.
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Homework

• Worksheet
• Quiz Monday on Boxplots outliers