1. Descriptive Statistics-IV (Measures of Variation)
QSCI 381 – Lecture 6
(Larson and Farber, Sect 2.4)

2. Deviation, Variance and Standard Deviation-I
The of a data entry xi in a population data set is the difference between xi and population mean , i.e.
The sum of the deviations over all entries is zero.
The is the sum of the squared deviations over all entries:
 is the Greek letter sigma.
Deviation
Population variance

3. Deviation, Variance and Standard Deviation-II
The is the square root of the population variance, i.e.:
Note: these quantities relate to the population and not a sample from the population.
Note: sometimes the standard deviation is referred to as the standard error.
Population standard deviation

4. The Sample variance and Standard Deviation
The and the of a data set with n entries are given by:
Sample variance
Sample standard deviation
Note the division by n -1 rather than N or n.

5. Calculating Standard Deviations
Step
Population
Sample
Find the mean
Find the deviation for each entry
Square each deviation
Add to get the sum of squares (SSx)
Divide by N or (n -1) to get the variance
Take the square root to get the standard deviation

6. Example
Find the standard deviation of the following bowhead lengths (in m):
(8.5, 8.4, 13.8, 9.3, 9.7)
Key question (before doing anything) – is this a sample or a population?

7. Formulae in EXCEL Calculating Means: Average(“A1:A10”)
Calculating Standard deviations: Stdev(“A1:A10”) – this calculates the sample and not the population standard deviation!

8. Standard Deviations-I
SD=0
SD=2.1
SD=5.3

9. Standard Deviations-II (Symmetric Bell-shaped distributions)
k = 2: proportion > 75%
k = 3: proportion > 88%
Chebychev’s Theorem:
The proportion of the data lying
within k standard deviations (k >1)
of the mean is at least 1 - 1/k2
68%
34%
95%
13.5%
99.7%

10. Standard Deviations-III (Grouped data)
The standard deviation of a frequency distribution is:
Note: where the frequency distribution consists of bins that are ranges, xi should be the midpoint of bin i (be careful of the first and last bins).

11. Standard Deviations-IV (The shortcut formula)

12. The Coefficient of Variation
The is the standard deviation divided by the mean - often expressed as a percentage.
The coefficient of variation is dimensionless and can be used to compare among data sets based on different units.
coefficient of variation

13. Z-Scores
The is calculated using the equation:
Standard (or Z) score

14. Outliers-I
Outliers can lead to mis-interpretation of results. They can arise because of data errors (typing measurements in cm rather than in m) or because of unusual events.
There are several rules for identifying outliers:
Outliers: < Q2-6(Q2-Q1); > Q2+6(Q3-Q2)
Strays: < Q2-3(Q2-Q1); > Q2+3(Q3-Q2)

15. Outliers-II
Strays and outliers should be indicated on box and whisker plots:
Consider the data set of bowhead lengths, except that a length of 1 is added! 5 10 15
Length (m)

16. Review of Symbols in this Lecture

17. Summary
We use descriptive statistics to “get a feel for the data” (also called “exploratory data analysis”). In general, we are using statistics from the sample to learn something about the population.