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Presentation Transcript

The Plan for Today

- Introduce descriptive statistics for variability:
- The range;
- The interquartile range;
- The population variance and standard deviation.
- The sample variance and standard deviation.

- Discuss their association with particular measures of central tendency.
- Discuss the meaning of the standard deviation.

Choosing a measure of variability

- When we discussed central tendency, we spoke of different reasons for choosing one descriptive statistic or another.
- Once the choice of central tendency has been made, the measure of variability is pretty much determined.
- Median interquartile range.
- Mean standard deviation.

The range

- Simple definition: range = largest value – smallest value.
- More complex definition: range = upper real limit of largest value – lower real limit of smallest value.
- Unless we are working from grouped data, we will use the simple definition.
- Example: the range of the Peabody scores we worked with last time is 100 – 57 = 43.

The range (cont.)

- Advantage of the range: it is very simple (even in its more complex form).
- Disadvantage of the range: it is determined entirely by two data points, and hence is extremely unstable.
- Hence, in general, don’t use it.

The interquartile range

- IQR = Q3 – Q1.
- Q3 = 75th percentile = median of the upper half of the data.
- Q1 = 25th percentile = median of the lower half of the data.
- Dealing with odd numbered data sets:
include the middle value in both halves.

- Example.

Final exam scores (N = 156)

16 17 20 20 22 23 25 27 28 28 28 29 29 30 30 31

31 31 31 32 32 33 33 33 34 34 35 35 35 35 36 36

36 36 36 36 37 37 37 37 37 37 38 38 38 38 38 39

39 39 39 39 40 40 40 40 40 40 40 41 41 41 41 41

41 41 42 42 42 42 42 42 42 42 42 42 42 43 43 43

43 43 43 43 43 44 44 45 45 45 45 45 45 45 45 45

45 45 46 46 46 46 46 46 47 47 47 47 47 48 48 48

48 48 48 48 48 48 48 48 49 49 49 49 49 49 49 49

49 49 50 50 50 50 50 50 51 51 51 51 52 52 52 52

52 52 52 52 52 53 54 54 54 54 55 55

Final exam scores (N = 156)

16 17 20 20 22 23 25 27 28 28 28 29 29 30 30 31

31 31 31 32 32 33 33 33 34 34 35 35 35 35 36 36

36 36 36 36 37 37 37 37 37 37 38 38 38 38 38 39

39 39 39 39 40 40 40 40 40 40 40 41 41 41 41 41

41 41 42 42 42 42 42 42 42 42 42 42 42 43 43 43

43 43 43 43 43 44 44 45 45 45 45 45 45 45 45 45

45 45 46 46 46 46 46 46 47 47 47 47 47 48 48 48

48 48 48 48 48 48 48 48 49 49 49 49 49 49 49 49

49 49 50 50 50 50 50 50 51 51 51 51 52 52 52 52

52 52 52 52 52 53 54 54 54 54 55 55

Final exam scores (N = 156)

16 17 20 20 22 23 25 27 28 28 28 29 29 30 30 31

31 31 31 32 32 33 33 33 34 34 35 35 35 35 36 36

36 36 36 36 37 37 37 37 37 37 38 38 38 38 38 39

39 39 39 39 40 40 40 40 40 40 40 41 41 41 41 41

41 41 42 42 42 42 42 42 42 42 42 42 42 43|43 43

43 43 43 43 43 44 44 45 45 45 45 45 45 45 45 45

45 45 46 46 46 46 46 46 47 47 47 47 47 48 48 48

48 48 48 48 48 48 48 48 49 49 49 49 49 49 49 49

49 49 50 50 50 50 50 50 51 51 51 51 52 52 52 52

52 52 52 52 52 53 54 54 54 54 55 55

The standard deviation

- A warning about conceptual understanding versus literalism.
- We’re going to start with some ways of talking about standard deviation that aren’t quite right, but they’ll help us grasp the concept.
- Then we’ll define it more accurately.
- Then we’ll see how to calculate it more efficiently.

The standard deviation (cont.)

- Just as the median (an order statistic) is associated with a measure of variability that involves order statistics…
- …the standard deviation measures variability using the concept of the mean.
- Conceptually, the standard deviation is like an average deviation from the mean.

Understanding the standard deviation

- We can easily calculate deviations from the mean.
- Problem: their mean is always zero.
- What’s an obvious, intuitive way to make them all positive?
- Absolute value leads to the mean absolute deviation.

Understanding the standard deviation (cont.)

- Problem: absolute values are very nasty to deal with in mathematics.
- So instead, we make the deviations positive by squaring them.
- The sum of these squared deviations from the mean is often called the “sum of squares.”

Conceptual definition of the sum of squares

The population variance

- The population variance is defined as the mean of these squared deviations from the mean.
- As a formula,

The population standard deviation

- The variance has a problem: it is not measured the same way as the data. The deviations were squared.
- We correct that problem by taking the square root of the population variance.
- The result is called the population standard deviation.

The population standard deviation

- Note that the words “standard deviation” evoke the idea we started with: an average or typical deviation from the mean.
- As a formula, the population standard deviation is

Variance and standard deviation of samples

- When we are characterizing the variability of a sample rather than a population…
- …the formula for the sum of squares is the same, but with the sample mean M in place of the population mean m.
- Instead of dividing by N we divide by (N – 1) to correct bias.

Variance and standard deviation of samples

- The sum of squares becomes
SS =

- The sample variance is s2 = SS/(N-1).
- The sample standard deviation is
s =

Peabody Picture Vocabulary Test

57 61 64 65 65 67 69 69 71 72

76 76 77 79 80 81 81 81 83 84

84 84 85 86 86 87 89 89 90 90

91 91 92 92 93 94 95 95 96 100

The computational formula

- Usually, the defining formula for the sum of squares is extremely cumbersome.
- A much more convenient formula is algebraically equivalent:

The computational formula

- This form is known as the computational formula for the sum of squares.
- You must know how to use the computational formula for exams.

Next time

- Review for the midterm.
- A study guide will be posted on the main class web page http://faculty.ucmerced.edu/jvevea/classes/010/psy010.html.