Review BPS chapter 1

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Review BPS chapter 1

• Picturing Distributions with Graphs
• What is Statistics ?
• Individuals and variables
• Two types of data categorical and quantitative
• Ways to chart categorical data bar graphs and
  pie charts
• Ways to chart quantitative data histograms and
  stem plots
• Interpreting histograms
• Time plots

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Example BPS chapter 1

• Indicate whether each of the following variables
  is categorical or
• quantitative.
• a. We have data on 20 individuals measuring
  amount of time it takes to
• climb five flights of stairs.
• b. During a clinical trial, an experimental pain
  relief drug is administered to
• individuals. Each individual is then asked
  whether s/he experienced
• any pain relief.

Quantitative
Categorical
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Objectives (BPS chapter 2)

• Describing distributions with numbers
• Measure of center mean and median
• Measure of spread quartiles and standard
  deviation
• The five-number summary and boxplots
• IQR and outliers
• Choosing among summary statistics

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Measure of center the mean
The mean or arithmetic average To calculate the
average, or mean, add all values, then divide by
the number of individuals. It is the center of
mass. Sum of heights is 1598.3 Divided by 25
women 63.9 inches
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Mathematical notation
Learn right away how to get the mean using your
calculators.
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Measure of center the median

• The median(M) is the midpoint of a
  distributionthe number such that half of the
  observations are smaller and half are larger.

1. Sort observations from smallest to largest.
2. Find the location of the median (L)
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Comparing the mean and the median

• The mean and the median are the same only if the
  distribution is symmetrical. In a skewed
  distribution, the mean is usually farther out in
  the long tail than is the median. The median is a
  measure of center that is resistant to skew and
  outliers. The mean is not.

Mean and median for a symmetric distribution
Mean Median
Mean and median for skewed distributions
Left skew
Right skew
Mean Median
Mean Median
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Mean and median of a distribution with outliers
Percent of people dying

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Impact of skewed data
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Example STAT 200 Midterm Score
Midterm 30 35 40 40 40 40 45 45 45 45 50 50 55 55
60 65 65 70 100 100
Descriptive Statistics Midterm Variable N
Mean StDev Minimum Q1 Median Q3
Maximum Midterm 20 53.75 18.98 30.00
40.00 47.50 63.75 100.00
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Measure of spread quartiles
The first quartile, Q1, is the value in the
sample that has 25 of the data at or below it.
The third quartile, Q3, is the value in the
sample that has 75 of the data at or below it.
Q1 first quartile 2.2
M median 3.4
Q3 third quartile 4.35
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Center and spread in boxplots
Largest max 6.1
Q3 third quartile 4.35
M median 3.4
Q1 first quartile 2.2
Five-number summary
Smallest min 0.6
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Boxplots for skewed data
Comparing box plots for a normal and a
right-skewed distribution
Boxplots remain true to the data and clearly
depict symmetry or skewness.
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IQR and outliers

• The interquartile range (IQR) is the distance
  between the first and third quartiles (the length
  of the box in the boxplot)
• IQR
  Q3 - Q1
• An outlier is an individual value that falls
  outside the overall pattern.
• How far outside the overall pattern does a value
  have to fall to be considered an outlier?
• The 1.5 X IQR Rules for Outliers

Low outlier any value lt Q1 1.5 IQR High
outlier any value gt Q3 1.5 IQR
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Example STAT 200 Midterm Score
Midterm 30 35 40 40 40 40 45 45 45 45 50 50 55 55
60 65 65 70 100 100

• IQR Q3 - Q1 63.75-40.0023.75

Low outlier any value lt Q1 1.5 IQR 40.00 -
1.5(23.75) 4.375 High outlier any value gt Q3
1.5 IQR 63.75 1.5(23.75) 99.375
Outliers !!
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Measure of spread standard deviation
The standard deviation is used to describe the
variation around the mean.
Mean 1 s.d.
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Calculations
Womens height (inches)
Mean 63.4 Sum of squared deviations from mean
85.2 Degrees freedom (df) (n - 1) 13 s2
variance 85.2/13 6.55 inches squared s
standard deviation v6.55 2.56 inches

• Well never calculate these by hand, so make sure
  you know how to get the standard deviation using
  your calculator.

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Choosing among summary statistics

• Because the mean is not resistant to outliers or
  skew, use it to describe distributions that are
  fairly symmetrical and dont have outliers. ?
  Plot the mean and use the standard deviation for
  error bars.
• Otherwise, use the median in the five-number
  summary, which can be plotted as a boxplot.

Box plot Mean s.d.
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Example 1

• Suppose a sample of twelve lab rats is found to
  have the following glucose levels
• 3 4 4 6 6 6 8 8 9 10 12 15
• 1. Find the five-number summary of the data and
  construct box-plot .
• 2. Based on the box plot, the data set is
• a. Skewed to left
• b. roughly symmetric
• c. skewed to right

Min3, Q15, M7, Q39.5, Max15

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Example 2
Suppose a researcher is recording fifty values in
a database. Suppose she records every value
correctly except the lowest value, which is
supposed to be 2 but which she incorrectly
types as 200. In the above scenario, the
effect of the researchers error on mean and
Median is a. Her calculated mean will be
lower than it would have been without the error,
but her calculated Median will remain unchanged.
b. Her calculated mean will be higher than it
would have been without the error, but her
calculated Median will remain unchanged. c. Her
calculated mean will remain unchanged, but her
calculated Median will be lower than it would
have been without the error. d. Her calculated
mean will remain unchanged, but her calculated
Median will be lower than it would have been
without the error.

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Example 2
In the above scenario, the effect of the
researchers error on standard deviation is
a. The error will not affect standard
deviation. b. Her calculated standard deviation
will be smaller than it would have been without
the error. c. Her calculated standard deviation
will be larger than it would have been without
the error. d. The error is likely to make the
calculated standard deviation negative.

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Example 3

• There are three children in a room -- ages 3, 4,
  and 5. If a four-year-old child enters the room,
  the
•  
• mean age and variance will stay the same.
• mean age and variance will increase.
• mean age will stay the same but the variance will
  increase.
• mean age will stay the same but the variance will
  decrease.
•