Transcript Measures of Dispersion

Measures of
Dispersion
Week 3
What is dispersion?
• Dispersion is how the data is spread out,
or dispersed from the mean.
• The smaller the dispersion values, the
more consistent the data.
• The larger the dispersion values, the more
spread out the data values are. This
means that the data is not as consistent.
Consider these sets of data:
• Grades from Test # 1 =
- 81,83,83,82,86,81,87,80,81,86
• Grades from Test # 2 =
- 95,74,65,90,87,97,60,81,99,76
• What differences do you see between the
two sets?
• What are the Mean scores? Ranges?
• Do you believe these grades tell a story?
Important Symbols to remember

= mean
• X = an individual value
• N = Population size
• n = sample population size
• i = 1st data value in population
Variance
• The average of the squares of
each difference of a data value
and the mean.
Standard Deviation
• is the measure of the average distance
between individual data points and their
mean.
• It is the square root of the variance.
• The lower case Greek letter sigma is used
to denote standard deviation.
 
How to Calculate Standard
Deviation
• Given the data set {5, 6, 8, 9}, calculate
the standard deviation.
• Step 1: find the mean of the data set
Sum of items x1  x2  x3  ...  xn
Mean 

Count
n
5  6  8  9 28
Mean 

7
4
4
How to Calculate Standard
Deviation
• Step 2: Find the difference between each
data point and the mean.
5  7  2
6  7  1
87 1
97  2
How to Calculate Standard
Deviation
• Step 3: Square the difference between
each data point and the mean.
 2   4
2
 1  1
2
1  1
2
 2  4
2
How to Calculate Standard
Deviation
• Step 4: Sum the squares of the
differences between each data point and
the mean.
4  1  1  4  10
How to Calculate Standard
Deviation
• Step 5: Take the square root of the sum
of the squares of the differences divided
by the total number of data points;
10
10 3.16227766


 1.58113883  
4
2
2
* The average distance between individual data
points and the mean is 1.58113883 units from 7
Standard Deviation
• Formula of what we just did:

n
1
2
(Xi  X )

n i 1
• For sample S.D. use 1/(n-1)
When to use Pop. vs. Sample
• When we have the actual entire population
(for example our class, 29 students), we
would use the Population formula.
• If the problem tells us to use a particular
formula; Pop. v. Samp.
• If we are working with less entire
population of a much larger group, we will
use the sample formula.
• (Which is one taken away from the pop. total)
Why is this useful?
• It provides clues as to how representative
the mean is of the individual data points.
• For example, consider the following two
data sets with the same means, but
different standard deviations.
Bowler # 1
{98, 99, 101, 102}
X  100
  1.58113883
Bowler # 2
{30 ,51, 149, 169}
X  100
  78.10889834
The mean with the standard deviation provides a better description of the data set.
TI-83 to Calculate Standard Deviation.
• Step 1. Press STAT,EDIT,1:EDIT
• Step 2. Enter your data in the L1 column,
pressing enter after every data entry.
• Step 3. Press STAT, CALC,1-Var stats
• Step 4. Scroll down to the lower case
symbol for the Greek letter sigma
• calculator help.
Let’s try one more by hand:
(1) Find the population standard deviation
for the following Stats class test grades:
78, 84, 88, 92, 68, 82, 92, 72, 88, 86, 76, 90
(a) How many grades fall within one SD of the mean?
(b) What percent fall within one SD of the mean?
* Now check it with the calculator!