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Psychology 10

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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.

- 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.

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- 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 =

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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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