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Class Session #2
Numerically Summarizing Data
• Measures of Central Tendency
• Measures of Dispersion
• Measures of Central Tendency and
Dispersion from Grouped Data
• Measures of Position
2
Recall the Definitions
• Parameter – a descriptive measure of
a population
(p = parameter = population,
usually in Greek letters)
• Statistic – a descriptive measure of a
sample
(s = statistic = sample,
usually in Roman letters)
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Common “descriptions”
• ? Average ? – “typical” as described in
the news reports
• Give some of today’s examples
• Data distributions’ “characteristics”
– Shape – look at a picture (histogram)
– Center – mean, mode, median
– Spread – range, variance, std. dev.
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Central Tendency Definitions
• Arithmetic mean – the sum of all the
values of the variable in the data set,
divided by the number of observations
• Population arithmetic mean computed using all the individuals in
the population (“mew” = μ) (≠ micro µ)
• Sample arithmetic mean – computed
using the sample data (“x-bar”)
x
• Note:
is a statistic, μ is a parameter
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More Central Tendency Defs
• Median – the value that lies in the
middle of the data, when arranged in
ascending order
(think of the median strip of highway in
the middle of the road)
• Mode – the most frequent observation
of the variable in the data set
(think “a la mode” in fashion /on top)
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Measures of Dispersion
Definitions
• Range (R) – the difference between the
largest data value (maximum) & the
smallest data value (minimum)
• Deviation about the mean – how
“spread out” the data is.
? for both population and sample variance,
the sum of all deviations about the mean
equals what ?
? the square of a non-zero number is ?
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More Measures of Dispersion
Definitions
• Population Variance – sum of squared
deviations about the population mean,
divided by the number of observations in
the population N (sigma squared)
• ? i.e. population variance is the mean of the
______ _________ ____ __ _________ ___ ?
Answer: Population variance is the mean of the
squared deviations about the population mean
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More Measures of Dispersion
Definitions
• Sample Variance – sum of the squared
deviations about the sample mean,
divided by the number of observations
minus one (s squared)
• Degrees of freedom is the “n-1”
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More Measures of Dispersion
Definitions
• Population Standard Deviation – the
square root of the population variance
(sigma, written as “σ”)
• Sample Standard Deviation – the
square root of the sample variance (s,
written as “s”)
BTW, later we discover “s” itself is a random
variable
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Empirical Rule for
Symmetric Data
• If the distribution is bell shaped:
68% of data within 1 std deviations
 95% of data within 2 std deviations
 99.7% of data within 3 standard deviations
of the mean

Rule holds for both samples & populations
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Supposing Grouped Data
• Approximate mean of a variable from
a frequency distribution
• Use the midpoint of each class
• Use the frequency of each class
• Use the number of classes
• Population Mean
• Sample Mean
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Supposing Grouped Data
• Weighted Mean
Good to use when certain data values
have higher importance (or weight)
[Sum of each value of variable times
its weight] / [sum of weights]
Examples of Grade Point Average
(GPA) and mixed nuts pricing
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Supposing Grouped Data
• Population Variance
sum of [(midpoint – mean)2 times
frequency] / [sum of frequencies]
• Sample Variance
as before except “-1” in denominator (the
degrees of freedom thing again)
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Supposing Grouped Data
• Population Standard Deviation
take square root of population variance
• Sample Standard Deviation
take square root of sample variance
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Measures of Position Definition
• z-Score – the distance that a data
value is from the mean in terms of
standard deviations. Equals (data
value minus mean) divided by
standard deviation]
• Population z-score
• Sample z-score
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Measures of Position
Definitions
• z-score equals [(data value minus
mean) divided by standard deviation]
• Is a "unitless" measure
• Can be “normalized” to get
• Mean of zero
• Standard Deviation of one
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Measures of Position Definitions
• z-score purpose is to provide a way
to "compare apples and oranges"
• by converting variables with
different centers and/or spreads
• to variables with the same center
(0) and spread (1).
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Measures of Position Definition
• Percentiles – k th percentile is a set of
data divides the lower k% from the
upper (1-k)%
• Divide into 100 parts, so 99 percentiles
exist
• “P sub k”
• Use to give relative standing of the data
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Measures of Position Definition
• Quartiles – divides the data into four
equal parts
•
•
•
•
•
Four parts, so three percentiles exist
“Q sub one, two, or three”
Q2 is the median of the data
Q1 is the median of the lower half
Q3 is the median of the upper half
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Numerical summary of data
• Five number summaries
• Interquartile range (Q3 – Q1) is
resistant to extreme values
• Compute five number summary
• Min value | Q1 | M | Q3 | max value
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Building a Box Plot – part 1
• 1. Calculate interquartile range
(IQR)
• 2. Compute lower & upper fence
• Lower fence = Q1 – 1.5 (IQR)
• Upper fence = Q3 + 1.5 (IQR)
• 3. Draw scale then mark Q1 and Q3
• 4. Box in Q1 to Q3 then mark M
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Building a Box Plot – part 2
• 5. Temporarily mark fences with
brackets
• 6. Draw line from Q1 to smallest
value inside the lower fence and a
line from Q3 to largest value inside
the upper fence
• 7. Put * for all values outside of the
fences
• 8. Erase brackets
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Distribution based on Boxplot
• Symmetric
• median near center of box
• horizontal lines about same length
• Skewed Right / Positive Skew
• median towards left of box
• right line much longer than left line
• Skewed Left / Negative Skew
• median towards right of box
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Which measure best to
report?
• Symmetric distribution
• Mean
• Standard Deviation
• Skewed distribution
• Median
• Interquartile Range
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Self Quiz
• When can the mean and the median be about
equal?
• In the 2000 census conducted by the U.S.
Census Bureau, two average household
incomes were reported: $41,349 and $55,263.
One of these averages is the mean and the
other is the median. Which is which and why?
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Self Quiz
• The U.S. Department of Housing and Urban
Development (HUD) uses the median to
report the average price of a home in the
United States.
• Why do they do that?
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Self Quiz
• A histogram of a set of data indicates that the
distribution of the data is skewed right.
• Which measure of central tendency will be
larger, the mean or the median?
• Why?
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Self Quiz
• If a data set contains 10,000 values arranged
in increasing order, where is the median
located?
• Matching: (parameter; statistic)
• _____ is a descriptive measure of a
population
• _____ is a descriptive measure of a
sample.
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Self Quiz
• A data set will always have exactly one
mode. (true or false)
• If the number of observations, n, is
odd; then the median, M, is the
value calculated by the formula
M=(n+1)/2
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Self Quiz
• Find the Sample Mean:
20, 13, 4, 8, 10
• Find the Sample Mean:
83, 65, 91, 87, 84
• Find the Population Mean:
3, 6, 10, 12, 14
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Self Quiz
• The median for the given list of six data
values is 26.5.
•
7 , 12 , 21 ,
, 41 , 50
• What is the missing value?
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Self Quiz
• The following data represent the monthly cell
phone bill for the cell phone for six randomly
selected months.
• $35.34
$42.09
$39.43
• $38.93
$43.39
$49.26
• Compute the mean, median, and mode cell
phone bill.
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Self Quiz
• Heather and Bill go to the store to
purchase nuts, but can not decide
among peanuts, cashews, or
almonds. They agree to create a mix.
They bought 2.5 pounds of peanuts
for $1.30 per pound, 4 pounds of
cashews for $4.50 per pound, and 2
pounds of almonds for $3.75 per
pound. Determine the price per
pound of the mix.