Statistics
A statistic: is any number that describes a
characteristic of a sample’s scores on a measure.
Examples are but not limited to average (arithmetic
mean), median and mode, and mean absolute
deviation.
Mean
• The mean is referred to often as the average or as the arithmetic
mean.
• You may already be familiar with the procedure for finding mean.
Given a set of numbers you total them using addition and then
divide by the number of numbers in the set.
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For example give the following test scores
80,90,70 (out of 100)
The total of 80 + 90 + 70 = 240
240 is out of a possible 300
Divide 240 by 300 and you get an average of .8
.8 is equal to 8 tenths or 80 one hundredths
In other words the average test score is 80%
On the next slide however we are going to look into where mean
comes from.
Mean continued
• Below are the test scores from the previous slide
on a number line. With out using the procedure
find the average.
X X
X
• 0 10 20 30 40 50 60 70 80 90 100
Mean continued
• Now mentally move the X’s toward the middle.
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X
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X
X X
X
• 0 10 20 30 40 50 60 70 80 90 100
Mean continued
• All of the scores are now averaged or we have found
the mean! This is why mean is referred to as a
measure of central tendency. We are literally moving
two points by half the distance from either side of
the mean until we achieve the mean by using the
procedure of algorithm.
X
X
X
• 0 10 20 30 40 50 60 70 80 90 100
Mean Challenge
• Can you find the average of the following scores
with out using the algorithm?
X
X
X
X
X
X
X
• 0 10 20 30 40 50 60 70 80 90 100
Mean Challenge
• How about this one? What do you begin to
encounter?
X
X
X
X
X
X
• 0 10 20 30 40 50 60 70 80 90 100
Median
• The median in a set of data is the number or value that
is in the middle of the data and is not always the same
as the mean. If you have an even amount of values in
your data set you must split the difference between
the two values. If there are an odd amount of values in
the data set you find the value in the middle.
• Given:10 20 50 60 70 (odd number set)
• Given 50 60 70 80 (Even number so the median is
65.
• Visualize the data set on a number line
Mode
• The mode is the number or value in a data set
that occurs the most. It is possible to have
bimodal and multimodal sets of data.
• Given: 10 20 30 30 30 100 (30 is the mode)
• Given: 10 20 50 50 60 70 70 (Bimodal data set)
• Given: 10 10 20 20 3030 (Multimodal data set)
Mean Absolute Deviation
• The absolute deviation is essentially the
average distance of each data point from the
mean.
• To calculate it add up the distance of each
data point in the set from the mean then
divide by the number of data points to come
up with the standard deviation of the mean.
Mean absolute deviation
X
Average
• 0 10 20 30 40 50 60 70 80 90 100
X
• First find the average of the data set which in this case is 50.
• Now what is the distance between the average and each data point?
• In this case the score of zero is 50 points from the average and so is 100
• Therefore 50 +50 = 100 and there are 2 data points therefore the Mean
absolute deviation is 100 divided by 2 or 50.
Mean absolute deviation
• Why is it useful to find the absolute deviation?
• The answer is it tells us about the distribution of the
data. A mean absolute deviation that is very small
could indicate a picture that might look like this on
the left versus the one right
X
XX X
X
X
X X X
X
X X
X
X
0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100
Standard Deviation
• Standard Deviation allows one to look at
where scores fall in relation to the data. This is
related to many times but not always to the
standard bell curve distribution.
Standard Deviation
• To calculate standard deviation
• 1) Find the mean of your data
• 2) Find the distance between each data point
and the mean.
• 3) Square the differences
• 4)find the sum of the squares
• 5) Divide the sum of the squares by the data
points.
Penny example with the bell curve
• Using excel in class students will flip a coin 10
times and record the results. The data points
will be entered into excel and graphed. Watch
what happens!