Part II Sigma Freud

1
Part IISigma Freud Descriptive Statistics

• Chapter 3 ? ? ? ?
• Viva La Difference Understanding Variability

2
What you will learn in Chapter 3

• Variability is valuable as a descriptive tool
• Difference between variance standard deviation
• How to compute
• Range
• Inter-quartile Range
• Standard Deviation
• Variance

3
Why Variability is Important

• Variability
• how different scores are from one particular
  score
• Spread
• Dispersion
• What is the score of interest here?
• Ah ha!! Its the MEAN!!
• Sovariability is really a measure of how each
  score in a group of scores differs from the mean
  of that set of scores.

4
Measures of Variability

• Four types of variability that examine the amount
  of spread or dispersion in a group of scores
• Range
• Inter-quartile Range
• Standard Deviation
• Variance
• Typically report the average and the variability
  together to describe a distribution.

5
Computing the Range

• Range is the most general estimate of
  variability
• Two types
• Exclusive Range
• R h - l
• Inclusive Range
• R h l 1
• (Note R is the range, h is the highest score, l
  is the lowest score)

6
Measures of variation Range

• Range
• The difference between the highest and lowest
  numbers in a set of numbers.
• 2, 35, 77, 93, 120, 540
• 540 2 538

7
Measures of variation Range

• What is the range of
• 2, 3, 3, 3, 4, 5, 6, 6, 7, 9, 11, 13, 15, 15, 15,
  16
• 24, 57, 81, 96, 107, 152, 179, 211
• 1001, 1467, 1479, 1680, 1134

8
Interquartile range

• Difference between upper (third) and lower
  (first) quartiles
• Quartiles divide data into four equal groups
• Lower (first) quartile is 25th percentile
• Middle (second) quartile is 50th percentile and
  is the median
• Upper (third) quartile is 75th percentile

9
Calculating the interquartile range for high
temperatures
interquartile range 52 35 17
10
Stem and Leaf 0730 Q1 Fall 2010 (N22)

• 2349
• 303344555666677779
• 401
• Q1 .25 (22)5.5 data point round up to 6th data
  pointvalue of 33
• Q2 n1/223/211.5 avg of 11th and 12th data
  pt 35.5
• Q3 .75(22)16.5 round up to17th data point
• Value of 37

11
Interquartile range and outliers

• Value can be considered to be an outlier if it
  falls more than 1.5 times the interquartile range
  above the upper quartile or more than 1.5 times
  the range below the lower quartile
• Example for high temperatures
• Interquartile range is 17
• 1.5 times interquartile range is 25.5
• Outliers would be values
• Above 52 25.5 77.5 (none)
• Below 35 25.5 9.5 (none)

12
Review Steps to Quartiles, Interquartile
Range,and Checking for Outliers

• 1) Put values in ascending OR descending order
• 2) Multiply .25 (n) for Q1
• 3) Multiply .75 (n) for Q3
• 4) Q3 - Q1 IQR
• 5) Q1 1.5 (IQR) value below smallest value in
  data set
• 6) Q3 1.5 (IQR) value above largest value in
  data set

13
Lets practice Finding Outliers

• What is the median, Q1, Q3, range, and IQR for
  the following? Then check for outliers.
• 10, 25, 35, 65, 100, 255, 350, 395 (n8)
• 10, 65, 75, 99, 299 (n5)
• 5, 39, 45, 59, 64, 74 (n6)

14
Computing Standard Deviation

• Standard Deviation (SD) is the most frequently
  reported measure of variability
• SD average amount of variability in a set of
  scores
• What do these symbols represent?

15
Why n 1?

• The standard deviation is intended to be an
  estimate of the POPULATION standard deviation
• We want it to be an unbiased estimate
• Subtracting 1 from n artificially inflates the
  SDmaking it larger
• In other wordswe want to be conservative in
  our estimate of the population

16
Things to Remember

• Standard deviation is computed as the average
  distance from the mean
• The larger the standard deviation the greater the
  variability
• Like the meanstandard deviation is sensitive to
  extreme scores
• If s 0, then there is no variability among
  scoresthey must all be the same value.

17
Computing Variance

• Variance standard deviation squared
• Sowhat do these symbols represent? Does the
  formula look familiar?

18
Standard Deviation or Variance

• While the formulas are quite similarthe two are
  also quite different.
• Standard deviation is stated in original units
• Variance is stated in units that are squared
• Which do you think is easier to interpret???

19
Same mean, different standard deviation Sample
variance and Sample standard deviation
20,31,50,69,80
Each number x1 Mean Distance from Mean
20 50 -30
31 50 -19
50 50 0
69 50 19
80 50 30
20
Then square each distance from mean and add
together

• (-30)2 (-19)2 (0)2 (19)2 (30)2
• 900 361 0 361 900
• 2522
• Divide by N-1 (N5)
• 2522/4630.5 Sample Variance
• To find sample standard deviation, take square
  root of variance 25.11

21
Same mean, different standard deviation
39,44,50,56,61
Each number x1 Mean Distance from Mean
39 50 -11
44 50 -6
50 50 0
56 50 6
61 50 11
22
Which data set has more variability?

• (-11)2 (-6)2 (0)2 (11)2 (6)2
• 121 36 0 121 36
• 314
• Divide by N-1 gives us sample variance
• 314/478.5
• Square root of 78.5 gives us sample standard
  deviation8.86

23
Measures of variation Standard deviation

• How about a more user-friendly equation?

24
Using Excels VAR Function
25
Using the Computer to Compute Measures of
Variability
26
Glossary Terms to Know

• Variability
• Range
• Standard deviation
• Mean deviation
• Unbiased estimate
• Variance