Transcript German Shepherd puppy

Measures of Variance
Measures of variance or measures of spread
are ways to measure how much a collection
of data is spread out.
With a measure of variation, a single
number describes how the values vary in a
set.
𝑅𝐴 − 𝑛𝑔𝑒
Range is the difference between the maximum
value and minimum value in a set of data.
Example: Here is a list of your test scores from
the first semester. What is the range of your test
scores?
90, 75, 83, 85, 94, 78, and 85
90 − 75 = 15
Mean Absolute Deviation
Mean Absolute Deviation is an average of how far each
data point in a set is from the mean of the set of data.
In other words, it is the “average distance from the
average.”
To find the MAD, find the absolute value of the difference
of each data value and the mean.
Then find the average of these values.
Standard Deviation
Standard Deviation describes the spread of data
compared to the mean.
We use this symbol 𝜎, which is the greek letter
sigma to represent standard deviation.
To find 𝜎 (𝑠𝑖𝑔𝑚𝑎) we take the square root of
the variance.
2
Variance (𝜎 )
To find the variance we determine the average of
the squared differences from the mean.
This allows us to get a more accurate picture of how
far apart the data is spread. The larger the variance,
the larger the standard deviation is.
Let’s try an example!
Shoulder Height in mm
You and your friends have measured
the height of your dogs in millimeters.
Dogs
First we are going to find the range.
We take the largest value and subtract from it
the smallest value.
Range = 600-220 = 380
The dog heights have a range of 380 mm.
We want to determine the mean,
MAD, the variance and the standard
deviation.
The first thing we need to determine is the
mean.
𝑀𝑒𝑎𝑛 =
600+525+400+360+235+220
=
6
390
Now we calculate each dog’s
difference from the mean and take the
absolute value:
Name:
Period:
Course:
|600-390|
Unit:
Lesson:
Homework
|360-390|
|220-390|
Date:
|525-390|
|400-390|
|235-390|
To calculate the MAD, we will find the
absolute value of each difference and
then determine the average.
𝑀𝐴𝐷 =
210 +|−30|+|−180|+|10|+|135|+|−155|
6
210 + 30 + 180 + 10 + 135 + 155
=
6
720
=
6
= 120
The Mean Absolute Deviation is 120 mm.
Sometimes the MAD does not give us a clear enough picture when we are
comparing data, so we should also determine the variance and standard
deviation of the data.
2
To calculate the variance (𝜎 ) we take
each difference, square it, and then
average the result.
𝜎2 =
=
(210)2 +(10)2 +(135)2 + −30 2 + −180 2 +(−155)2
6
44,100+100+18,225+900+32,400+24,025
6
119,750
=
6
= 19,958.33
So, the variance is 19, 958.33
Now we can find the Standard
Deviation (𝜎)
To go from 𝜎 2 𝑡𝑜 𝜎 all we need to do is take the
square root.
So, 𝜎 =
19,958.33 = 141.27 …
or 141 (to the nearest mm)
This gives us a better understanding of the spread of data.
Now we can show which heights are
within one standard deviation (141
mm) of the mean.
141
141
Now, we can determine which dogs are extra large and which are extra small.
Notice that the German Shepherd and Chihuahua are well outside of the one
standard deviation from the mean.