Transcript Statistics Chapter 2 Exploring Distributions

Section 2.3
Measures of Center and Spread

Mean (average):
 Sum of x’s / total number of values
𝑥
𝑥=
𝑛
 It is the balance point of a distribution

Median is the value the divides the data into 2
halves.
 The middle number or average of middle 2 numbers.
 1,3,6,8,9
Median is 6
 1,3,7,9,10,15
Median is (7+9)/2=8
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An extreme outlier has a larger effect on the
mean than the median. Why?
 1,4,6,8 Find the mean and median
 1,4,6,8 and 55: Now find the mean and median
 Remember the mean is the balance point of a
distribution and the median is the middle number.
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Use the Speed of Mammals to investigate median and the 5-Number Summary:
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Speed (mi/h) Predator (1=Y, 0=N
30
1
30
1
70
1
30
0
39
1
40
0
25
0
45
0
42
1
32
0
20
0
48
0
40
0
50
1
11
0
35
0
12
0
40
0
Find the median of the Predators:
Find the lower or first quartile of the Predators: (omit the median)
Find the upper of third quartile of the Predators: (omit the median)
What are the minimum and maximum values for Predators:
The 5-Number Summary and gives a measure of center and spread.
Now do the same for the non-predators:
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Use the 5-Number Summary to create a box
and whisker plot.
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When comparing two groups use the same
scale and place box and whisker plots on top
of each other.
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Use the mammal speed data to compare wild
mammals to non-wild mammals with box and
whisker plots.
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Page 63 D16
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Page 69 P16, 18, 19, 21
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The IQR is a measure of the spread of the
middle 50% of the data.

IQR=Q3-Q1

This can be use to determine if a value is an
outlier.
 If the value is more than 1.5 times the IQR from
the nearest quartile.
 Outliers in a Boxplot.

Measuring Hand Span Activity
 Using a ruler, measure your hand span to the
nearest mm.
 Write your result on the board.
 Enter all data into calculator.
 Find the 5 Number Summary of the data.
▪ (1-var stats)
▪ Create a box plot
 Enter L1-mean into L2
 Use 1-var stats to find the sum of this:
x
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Think about what subtracting the mean (x)
from your data values does.
 It tells how far your hand span is from the class
average.
▪ + and your hand span is larger
▪ - and your hand span is smaller
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Now think about what summing all of your
differences does. What was the result?
 Differences from the mean will always sum to be Zero.
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Rapid Fire Example , Rapid Fire 2
A measure of the average difference from the mean gives
you a kind of level of consistency.
 Think of a target in an archery contest.
If you wanted to measure someone’s accuracy, how would you
suggest doing it?

The standard deviation of a data set is typically used when
the data is roughly normally distributed.
 It measures the “average” distance from the mean.
 This is a measure of spread. (how far are the arrows spread apart)

To calculate a SD, we must figure out a way to find the
average difference from the mean, but we know that the
sum of the differences is always = zero.
 So we square the difference, then sum that, then divide to find
average sum of squares, then take the square root.
s
 (x  x)
n 1
2
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The SD squared: (s2) is called the variance.
 It is not used as often, but is important to know
what it is, in case it is a piece of information given.
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You can get the SD from the calculator.
 Sx is the SD of a sample of data.
 σx is the SD of an entire population (we will
discuss the difference later)
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Note the other sums and values you can get.
 Σx will give the sum of you list of values
 Σx2 will give the sum of squared values
 n is your total number of data values in the list
 Then is your 5-Number Summary
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Frequency Tables show the values and the
frequency that they occur in the data set.
You can easily find the mean of this data:
(x  f )

x
n
 (x  x)
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You can find the SD:
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It is easy to find with Calculator:
s
2
f
n 1
 L1:x values, L2: frequencies, 1-var stat L1,L2
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Page 70 P22-P25
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Page 71: E29, 31, 33, 35, 37, 38, 43