Transcript Statistics intro

Statistics:
How We View the
World…
Day 1:
Center and Spread
One-Minute Question
• Find the mean of the
following grades:
• {70, 70, 80, 92, 98, 100}
One-Minute Question
• Arithmetic Mean = average =
(70+70+80+92+98+100)/6
• = 85
Review
• Find all the other “Measures
of Central Tendency” that
you know…
• {70, 70, 80, 92, 98, 100}
Review
• Did you name…
• Median = 86
• Mode = 70
• {70, 70, 80, 92, 98, 100}
Review
• Besides “Central
Tendency”, what else do we
care about?
Review
• Find all the “Measures of
Dispersion” that you know…
• {70, 70, 80, 92, 98, 100}
Review
Range = 30
• Mean Deviation = 11.67
• Inner-Quartile Range (IQR)
= 28
• {70, 70, 80, 92, 98, 100}
New Concept
To Find Standard Deviation
• 1. Find each number’s difference
from the mean.
• 2. Square these differences.
• 3. Find the average of these squared
differences.
• 4. Take the square root of your
result.
•
{70, 70, 80, 92, 98, 100}
Standard Deviation
• In other words:
• Standard Deviation = σ
x





n
• σ=
i1
2
i
n
• {70, 70, 80, 92, 98, 100}

Standard Deviation
• In other words:
• Standard Deviation =
15  15  5  7  13  15
6
 12.37
2
2
2
2
2
• {70, 70, 80, 92, 98, 100}
2
Variance = (Standard Deviation)2
  x   
n
Variance 
i1
in this case...
2
i

n
15  15  5  7  13  15
 153
6
2
2
2
2
2
2
So, Who Cares??
• All of us who are “Normal”!
So, Who Cares??
• Remember that histograms
are graphs of the
distribution of data???
• Well, think about other –
more general distributions.
• Suppose we roll a single die
1000000 times.
• What should the distribution
of number of dots on the top
face look like?
• Uniform distribution
• (Close to constant!)
1
2
3 4 5
6
Frequency
So, Who Cares??
So, Who Cares??
Frequency
• What should the distribution
be for a VERY, VERY easy
test?
• It should be skewed to the left.
• (There should be lots of data
to the right of the graph.)
F
D
C B A
So, Who Cares??
Frequency
• But the heights of 20 year
old males is normal.
• That means that the data
forms a bell-shape that is
symmetric about the mean.
Furthermore, the percentage of the
area covered by each standard
deviation from the mean is shown by
this graph.
So, suppose the mean of a normal
distribution is 100 with a standard
deviation of 15. What percentage of
the scores lie between 85 and 115?
So, suppose the mean of a normal
distribution is 100 with a standard
deviation of 15. What percentage of
the scores lie between 70 and 115?
In order to find how many standard
deviations away an x value is from
the mean we use a z-score.
x
z(x) 

If the mean of a set of test grades is
84 with a standard deviation of 6,
what is the z-score for a test grade
of 96?
What percent
of the other
students
scored lower
that a 96?
For homework,
work all the odd problems on pages
262 and 266