Download Statistics Chapter 2 Exploring Distributions

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Section 2.4
Working with Summary Statistics

Measures of Center
 Mean, Median, and sometimes Mode

Measures of Spread
 Standard Deviation, and Quartiles (Q1 and Q3)
 Remember SD gives an “average” deviation from
the mean.
 The quartiles divide the data into 25% portions.

Lets say we know the mean value of the
homes in a community along with the total
number of homes: $213,500; 412 homes

We also know the tax rate: 1.5%

How can we use the mean to determine the
total tax dollars received by the community?
 The mean is used to represent the value of every home.

When we describe the center of the annual
income of a group of people, it is typical to use
the median instead of mean…Why?

There are typically a large group of people
clustered around the low end of the scale with a
few having very large incomes. This creates a
distribution that is…..
 Skewed right and therefore the mean gives a measure
that is higher than expected. The median filters out
these extreme values.


Create a dot plot of the following data:
City
Country
Temperature (F)
Addis Ababa
Algiers
Bangkok
Madrid
Nairobi
Sao Paulo
Warsaw
Ethiopia
Algeria
Thailand
Spain
Kenya
Brazil
Poland
32
32
50
14
41
32
-22
Now create a dot plot of the distance the
temperature is from freezing (32o). Positive if
above freezing, negative if below.

Recentering a set of data is when we add or
subtract a constant from each data value.
 This shifts the data on the number scale, but does
nothing to change the shape or spread.
 The mean will be shifted by the constant added or
subtracted.

Now use the same data and convert it to celsius.
 Simply multiply the degrees above or below freezing
by 1/1.8.

What happened to your data set?
 Shape
 Mean?
 Standard deviation?


Notice that the mean and SD are multiplied by
1/1.8, but the shape stays the same.
This simply shrinks or if by a number greater
than 1, stretches the distribution.

A summary statistic is resistant to outliers if it is
not changed very much when the outlier is
removed from the data.

A summary statistic is sensitive to outliers if it is
changed significantly when the outlier is
removed from the data.

Remember our discussion of Mean vs Median

Refer to page 77: example of television viewers

Percentile: If a value is at the kth percentile, then k% of
the data is at or lower than this value.

Example: You got a 32 on math portion of the ACT. You
are told this is the 86th percentile.
 That means 86% of the test takers scored at a 32 or
lower.
 It also means that 14% scored above a 32.

This is a measure of where a data value lies
within the data set.


Frequency plot where the plotted points
show you the accumulated percent of data up
to that point.
Example: page 78

Page 80 E47, 49, 53 - 56

Re-Centering happens when you…..??
 What happens to the shape?
 What happens to the center?
 What happens to the spread?

Re-Scaling happens when you….??
 What happens to the shape?
 What happens to the center?
 What happens to the spread?