Transcript The Average - Department of Mathematics

Average, Median,
and Standard Deviation
Center and Spread
• The average and the median are measures of
the center.
• The standard deviation is a measure of the
spread.
Center and Spread
The centers of the two histograms are the same,
but the histogram on the right is more spread out.
Average, Mean, Median
• The average of a list of numbers equals
their sum, divided by how many there are.
• The average is sometimes called the (arithmetic)
mean.
How Many Are
Above Average?
What Happens to the Mean?
The Median
•
The median of a list is the value with half the entries
to the left, and half the entries to the right.
•
Find the median of 9, 1, 2, 2, 0:
Order the list: 0, 1, 2, 2, 9
Find the median of 7, 2, 1, 5:
Order the list: 1, 2, 5, 7.
The average of 2 and 5 is 3.5
•
Average vs. Median Income
• For persons age 25 and over in the U.S.
would the average or the median be higher
for income? For years of schooling
completed?
The Tail of a Histogram
Average is bigger
than the median
Average is about the
same as the median
Average is smaller
than the median
Standard Deviation
The standard deviation tries to quantify the
spread.
The root-mean-square
• The average of the list 0, 5, -8, 7, -3 is 0.2.
• The positive numbers ‘wipe out’ the
negative ones.
• We could neglect the signs. The average of
0, 5, 8, 7, 3 is 4.6
• Instead, we use the root-mean-square.
The root-mean-square
1. Square all the entries. This will get rid of
all the signs.
2. Take the mean (average) of the squares.
3. Take the square root of the mean.
Example
•
Find the root-mean-square of
0, 5, -8, 7, -3.
1. Square: 0, 25, 64, 49, 9.
2. Mean: (0 + 25 + 64 + 49 + 9) : 5 = 29.4
3. Root:
29 . 4  5 . 4
Computing the
Standard Deviation
Find the standard deviation of 20, 10, 15, 15.
• Find the average: 15.
• Find the deviations from the average:
5, -5, 0, 0.
• Find the root-mean-average:
2
2
2
5  ( 5 )  0  0
4
2

25  25  0  0
4
 3 .5
Which list has the larger SD?
• 50, 40, 60, 30, 70, 25, 75
• 50, 40, 60, 30, 70, 25, 75, 50, 50, 50
68 – 95 - 99
• In many cases, roughly 68 percent of the
entries on a list are within 1 SD of average;
roughly 95% within 2 SDs; 99 percent are
within 3 SDs of the average.
Example
• Average height for boys age 11 is 146 cm
and the SD was 8 cm.
• How much below average is a boy 170 cm
tall? 148 cm tall?
• How tall is a boy 1.5 SD below average?
• Four boys are 150 cm, 130 cm, 165 cm, 140
cm. Who is unusually short, about average,
unusually tall?