Chapter 1 Section 2
Room: 2210
Class: AP Statistics
Chapter 1 Section 2
Measuring Center Through Mean & Median
The Sample Mean, symbol , read as “x bar” OR The Population Mean,
symbol , read as “mu”
Add their values and divide by the number of observations
Chapter 1 Section 2 Continued…
Measuring Center Through Mean & Median Continued…
The Median, symbol M, formal version of midpoint with a specific rule of
calculation
Half of the observations are smaller and the other half are larger Step 1: Arrange all observations in order of size, from smallest to
largest
Step 2 Version 1: If the number of observations n is odd, the
median M is the center observation in the ordered list
Step 2 Version 2: If the number of observations n is even, the
Chapter 1 Section 2 Continued…
Measuring Center Through Mean & Median Continued…
Median is unlike mean, in that median is resistant
Median counts the data value as an observation not a value Mean uses the data value as an actual number
The mean and median of a symmetric distribution are close together.
Chapter 1 Section 2
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Measuring Spread Through The Quartiles
Range ~ difference between the largest and the smallest observations
Quartiles
Mark out the middle half
Arrange the observations in increasing order and locate the median
M in the ordered list of observations
First quartile, Or Q1 : lies one quarter of the way up the list of
observations, larger than 25% of the observations
Chapter 1 Section 2
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Measuring Spread Through The Quartiles Continued…
Second Quartile Or Median: larger than 50% of the observations
Third Quartile, Or Q1: lies three-quarters of the way up the list of observations, larger than 75% of the observations
Is the median of the observations whose position in the ordered list is to
the right of the location of the overall median.
Interquartile Range, Or IQR: Gives the range covered by the middle half of the data
Is the distance between the first and the third quartiles
The IQR is the basis of a rule of thumb for identifying suspected
outliers.
Chapter 1 Section 1
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When talking about the overall pattern of a distribution, remember
“C.C.S.S.O” , Must ALWAYS include the following
Context ~ who, what, and why????
Center
Spread ~ how spread out is the data
Shape ~ distributions come in a limitless variety of shapes, but certain
shapes arise often enough to have their own names.
Bell-Shaped…
Symmetric (is not always bell-shaped)- one half is roughly a mirror
image of the other
Skewed right ~ the right side of the distribution extends much farther out than the left side
Skewed left ~ the left side of the distribution extends much farther out than the right side
Uniform ~ the variables have approximately equally likely outcomes
Outliers ~ observations which differ markedly from the pattern
Chapter 1 Section 1
Continued…
Also, peaks or clusters that indicate the data fall into natural
subgroups.
Also, Granularity – values occur only at fixed intervals ( such as multiples of 5 or 10)
Chapter 1 Section 2
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5 Number Summary
What is the “Five Number Summary?” Minimum
Quartile One Median
Third Quartile Maximum
Will Always be given and received in the following way!:
Chapter 1 Section 2
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Boxplots (Modified and Original)
Boxplots:
Best used for side-by-side comparison of more than one distribution Can be drawn horizontally or vertically
Include numerical scale in graph Label axes and title graph
When viewing a boxplot: 1st locate the median
2nd look at the spread
Boxplots indicate the symmetry and skewness of a distribution
Symmetric, the first and the third quartiles are equally distant from
the median
Chapter 1 Section 2
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Modified Box Plot
Modified Boxplot, same as original boxplot, however it plots outliers as
isolated points
From here on out when I say boxplot, I mean modified boxplot Graph of the “Five Number Summary”
Modified described
A central box spans the quartiles
A line in the box marks the median M
Observations more than 1.5 x IQR outside the central box are plotted
individually
Lines extend from the box out to the smallest and largest
observations that are not outliers
Chapter 1 Section 2
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Measuring Spread Through Standard Deviation
Standard Deviation measures spread by looking at how far the
observations are from their mean
Standard deviation of a Sample Standard deviation of a Population
Chapter 1 Section 2
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Chapter 1 Section 2
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Measuring Spread Through Variance
Variance ~ set of observations is the average of the squares of the
deviations of the observations from their mean
Chapter 1 Section 2
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Degrees Of Freedom: the numbers n-1
Properties of Standard Deviation
s or σ , measures spread about the mean and should be used only when the
mean is chosen as the measure of center
s = 0 or σ = 0 , only when there is no spread, this happens only when all
observations have the same value. Otherwise the standard deviation is greater than 0, as observations become more spread out about the mean, standard deviation becomes larger.
s or σ , like mean are not resistant. Strong skewness or a few outliers can
make the standard deviation very large
What do I choose ??
The “Five Number Summary” is usually better than the mean and standard
deviation for describing a skewed distribution or a distribution with strong outliers. Use mean and standard deviation for reasonably symmetric
Chapter 1 Section 2
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Linear Transformation
Does not change the shape of a distribution, however changing the units of
measurement can affect the center and spread of the distribution
Rules of Linear Transformation
Multiplying each observation by a positive number b multiplies both measures
of center (mean and median) and measures of spread ( standard deviation and IQR) by b
Adding the same number a (either positive or negative) to each observation
adds a to measures of center and to quartiles but does not change measures of spread.
Adding the constant a shifts all the values of x upward or downward by the same
amount