1. Measures of Variation
Skill 11

2. Objectives Find the range Find the sample and population variance
Find the sample and population standard deviation.

3. The range is one measure of variation.
The range is the difference between the largest and the smallest values of data.

4. Example–Range
A large bakery regularly orders cartons of Maine blueberries. The average weight of the cartons is supposed to be 22 ounces. Random samples of cartons from two suppliers were weighed. The weights in ounces of the cartons were
Supplier I:
Supplier II:
Find the range and mean.

5. Supplier I Supplier II Range: Range: 27-17 = 10 27-17 = 10 Mean: Mean:
= 22
= 22
Median:
Median:
= 22
= 20
Mode:
Mode:
= 22
= 27

6. Variance and Standard Deviation
We need a measure of the distribution or spread of data around an expected value (either 𝑥 or  ). Variance and standard deviation provide such measures.

7. Sample Standard Deviation
Variance and Standard Deviation
Sample Standard Deviation
Sample Variance
𝑠 2 = (𝑥− 𝑥 ) 2 𝑛−1
𝑠= (𝑥− 𝑥 ) 2 𝑛−1

8. Variance and Standard Deviation
In statistics, the sample standard deviation and sample variance are used to describe the spread of data about the mean x .

9. Find the sample variance and sample standard deviation.
Example –Sample Standard Deviation (Defining Formula)
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.
Find the sample variance and sample standard deviation.

10. 𝑠 2 = 70 5 =𝟏𝟒 ∴𝑠= 𝑠 2 = 14 ≈𝟑.𝟕𝟒 6 𝑛= 36 6 =6 𝑥 = 𝑠 2 = (𝑥− 𝑥 ) 2 𝑛−1
𝑠 2 = (𝑥− 𝑥 ) 2 𝑛−1 𝑛= 6
36 6 =6
𝑥 =
𝑠 2 = (𝑥−6) 2 6−1
𝑠 2 = (2−6) 2 + (3−6) 2 + (3−6) 2 + (8−6) 2 + (10−6) 2 + (10−6) 2 6−1
𝑠 2 = (−4) 2 + (−3) 2 + (−3) 2 + (2) 2 + (4) 2 + (4) 2 5
𝑠 2 =
𝑠 2 = 70 5 =𝟏𝟒
∴𝑠= 𝑠 2 = 14 ≈𝟑.𝟕𝟒

11. Variance and Standard Deviation

12. Example–Variation
A store has a very limited selection of spinners. In fact, the Trading Post has only eight different types of spinners for sale. The prices (in dollars) are
Since the Trading Post has only eight different kinds of spinners for sale, we consider the eight data values to be the population.
Find the population variance and population standard deviation.

13. 𝜎 2 = (𝑥−𝜇) 2 𝑁 N= 8
=2.14 𝜇=
𝜎 2 = (𝑥−2.14) 2 6−1 = =
𝜎 2 = =𝟎.𝟎𝟒𝟕
∴𝜎= 𝜎 2 = ≈𝟎.𝟐𝟏𝟕

14. Example–Variation
Dr. Packer is the faculty sponsor for a student volunteer program. For several years, Dr. Packer has kept a careful record of the total number of hours worked by students in the program and the hours they volunteered is listed below.
Find the mode, median, mean, range, IQR, Outliers, variance and standard deviation.

15. d) Range.
a) Mode.
50−14= 𝟑𝟔 23
b) Median.
e) IQR.
32−21= 𝟏𝟏 23
c) Mean.
f) Outliers. 7
=16.5 n= =
𝟒𝟖.𝟓
190 7 =𝟐𝟕.𝟏𝟒
𝑥 =
21−16.5=
𝟒.𝟓
𝑶𝒖𝒕𝒍𝒊𝒆𝒓; 𝟓𝟎

16. 𝑠 2 = (𝑥− 𝑥 ) 2 𝑛−1
= (𝑥−27.14) 2 7−1
𝑠 2 =
𝑠 2 =
𝑠 2 =𝟏𝟑𝟏.𝟖𝟏
∴𝑠= 𝑠 2 = ≈𝟏𝟏.𝟒𝟖

17. 11: Measures of Variation
Summarize Notes
Questions?
Homework
Worksheet