1. 3.2 Measures of Variation
CHAPTER 3
AVERAGES AND VARIATION

2. Measures of Variation
 An average is an attempt to summarize a set of data
using just one number.
 an average taken by itself may not always be very meaningful.
 We need a statistical cross-reference that measures
the spread of the data.

3. Measures of Variation
 Two sets could have the
same mean and look very
different in terms of
spread.
Example:  
Set A: 10, 10, 11, 12, 12 
Set B: 2, 4, 11, 18, 20     
Both have a mean of 11.

4. The Range
 The range is a measure of variation, it is the
difference between the largest and smallest values of
a data distribution.
 Does not tell how much other values vary from one another or
from the mean
 Range = highest value – lowest value

5. Measuring Spread – Deriving the Formulas


6. Measuring Spread – Deriving the Formulas


7. Variance and Standard Deviation
 Standard deviation and variance are statistical
measures of spread used to assess how far away
individual data points are from the mean, or average,
within a set of data.
 Knowing how far each data point lies from the average allows
you to make more accurate conclusions about a data set,
population or sample, because you can pick out outliers and
extreme values and note how they affect your results.

8. Conceptual Difference Between Standard
Deviation and Variance
 The variance of a data set measures the mathematical
dispersion/spread of the data relative to the mean. However,
though this value is theoretically correct, it is difficult to apply
in a real-world sense because the values used to calculate it
were squared.
 The standard deviation, as the square root of the variance
gives a value that is in the same units as the original values,
which makes it much easier to work with and easier to
interpret.
  Thus the standard deviation is the measure of spread that most
people refer to.
 The standard deviation just tells us, on average, how far each
individual is from the actual mean (in the same units of measure of our
data).

9. Variance and Standard Deviation
 The formulas for variance and standard deviation
differ slightly, depending on whether we are using a
sample or the entire population.

10. Defining Formulas
 Sample Variance
Hint: When using
these formulas, it helps
to use a chart for the
intermediate work.
 Sample Standard Deviation
AP Formula Sheet:
To find variance: square the result

11. Rounding Note: Rounding Errors are going to happen!
If you do your calculations by hand, and reenter
intermediate values into a calculator, carry two more
digits than occur in the original data. If your
resulting answers vary slightly from those in the text, do
not panic. All text answers are computer/calculator
generated.

12. Page 96
Example 6 – Sample Standard Deviation
Big Blossom Greenhouse was commissioned to
develop an extra large rose for the Rose Bowl Parade.
A random sample of blossoms from Hybrid A bushes
yielded the following diameters (in inches) for
mature peak blooms.
2 3 3 8 10 10
Use the defining formula to find the sample variance
and standard deviation.

13. Solution– Sample Standard Deviation
x x–x (x – x)2
2
3
3
8
10
10
∑x ∑(x – x)2

14. Population Variance and
Population Standard Deviation
 In most applications of statistics, we work with a
random sample of data rather than the entire
population of all possible data values.
 If we have data for the entire population, we can
compute the population mean , population
variance 2, and population standard deviation
(lowercase Greek letter sigma)

15. Population Variance and
Population Standard Deviation
 Population Mean
 Population Variance
 Population Standard Deviation

16. Using the Calculator
1. Enter the data in L1
2. Hit STAT, tab over to CALC, choose 1: 1-VarStats
3. Hit 2nd 1 ENTER To find the Variance:
After 1-Var Stats
Screen
1. Hit VARS
2. Choose option 5:
Sample Standard Statistics
Deviation 3. Choose the
Standard
Population Standard Deviation
Deviation 4. Hit the “squared”
Key, ENTER

17. Page 101
Chebyshev’s Theorem
 The concept of data spread about the mean can be
expressed quite generally for all data distributions
(skewed, symmetric, or other shapes) by using the
remarkable theorem of Chebyshev.
Chebyshev’s Theorem
For any set of data (either population or sample) and
for any constant k, greater than 1, the proportion of the
data that must lie within k standard deviations on either
side of the mean is at least

18. Page 102
Chebyshev’s Theorem
Results of Chebyshev’s Theorem
 For any set of data
 At least 75% of the data fall in the interval from μ – 2σ to μ + 2σ
 At least 88.9% of the data fall in the interval from μ – 3σ to μ + 3σ
 At least 93.8% of the data fall in the interval from μ – 4σ to μ + 4σ
 Notice that Chebyshev’s theorem refers to the
minimum percentage of data that must fall within the
specified number of standard deviations of the mean.
 If the distribution is mound-shaped, an even greater
percentage of data will fall into the specified intervals.
(Chapter 6)

19. Page 102
Example 8 – Chebyshev’s Theorem
Students Who Care is a student volunteer program in
which college students donate work time to various
community projects such as planting trees. Professor Gill
is the faculty sponsor for this student volunteer program.
For several years, Dr. Gill has kept a careful record of x =
total number of work hours volunteered by a student in
the program each semester. For a random sample of
students in the program, the mean number of hours was
x = 29.1 hours each semester, with a standard deviation s
= 1.7 of hours each semester.
Find an interval A to B for the number of hours
volunteered into which at least 75% of the students in
this program would fit.

20. Homework
 Page 104
 #1 – 4
 #5 (by hand)
 #10, 11, 12, 13
 #17 & 18 (a, b, c)