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# Psychology 10 - PowerPoint PPT Presentation

Psychology 10. Analysis of Psychological Data February 5, 2014. 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.

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### Psychology 10

Analysis of Psychological Data

February 5, 2014

• 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.

• 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.

• 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.

• 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

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

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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

• 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.

• 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.

• 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.

• 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.”

• The population variance is defined as the mean of these squared deviations from the mean.

• As a formula,

• 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.

• 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

• 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.

• The sum of squares becomes

SS =

• The sample variance is s2 = SS/(N-1).

• The sample standard deviation is

s =

57 61 64 65 65 67 69 69 71 72

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84 84 85 86 86 87 89 89 90 90

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• Usually, the defining formula for the sum of squares is extremely cumbersome.

• A much more convenient formula is algebraically equivalent:

• This form is known as the computational formula for the sum of squares.

• You must know how to use the computational formula for exams.

• 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.

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70 72 74 77 80 82 84 87 88 95

97 100