# Statistical Foundations: Measures of Location and Central Tendency and Summation and Expectation

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1 Statistical Foundations: and Central Tendency and and Lecture 4 September 5, 2006 Psychology 790 Lecture #4-9/05/2006 Slide 1 of 26

2 Today s Lecture Today s Lecture Where this Fits central tendency/location (Hays, Chapter ). (Hays, Appendix A). Algebra of s (Hays, Appendix B). Lecture #4-9/05/2006 Slide 2 of 26

3 The Big Picture Today s Lecture Where this Fits You probably have already seen the mean, median, and mode in action. We will use many of these very frequently. is something that will play a big role in the weeks to come. We will use expectation to form our variance. Regression lines are formed by taking the expectation (or mean) of a conditional distribution of the Dependent Variable (Y ) given the independent variables (X). Regression lines fit with least squares always include the mean of X and the mean of Y on the line. Regression lines fit with the minimum absolute deviation always include the median of X and the median of Y on the line. Lecture #4-9/05/2006 Slide 3 of 26

4 The summation operator is defined as follows: Properties Double Bonus Coverage X i = X 1 + X X n Example with data X 1 = 3 X 2 = 5 X 3 = 4 3 X i = = 12 You should remember the use of this operator from the well known statistic, the mean. Ȳ = n X i n = 3 X i 3 = = 12 3 = 4 Lecture #4-9/05/2006 Slide 4 of 26

5 Sum of a Constant of a constant k: Properties Double Bonus Coverage k = nk Example: k = 4, n = 3 k = = 3 4 = 12 Lecture #4-9/05/2006 Slide 5 of 26

6 Sum of a Sum Sum of a sum: Properties Double Bonus Coverage [X i + Z i ] = X i + Example: X 1 = 3 X 2 = 5 X 3 = 4, Z 1 = 1 Z 2 = 8 Z 3 = 2 [X i + Z i ] = X i + Z i = = 23 Without the property: X 1 +Z 1 = 4,X 2 +Z 2 = 13, X 3 +Z 3 = 6 Z i [X i + Z i ] = = 23 Lecture #4-9/05/2006 Slide 6 of 26

7 Sum of a Linear Combination of a Variable For constants a and c: Properties Double Bonus Coverage [a + cy i ] = na + c X i Think of this as a distribution of the Sum : [a + cy i ] = [a] + [cy i ] = na + c X i Example: X 1 = 3 X 2 = 5 X 3 = 4, a = 2, c = 5, n = 3 [a + cy i ] = X i = = 66 Lecture #4-9/05/2006 Slide 7 of 26

8 Double Operator The double summation operator is defined as: Properties Double Bonus Coverage m X ij = j=1 (X i1 + X i X im ) = X 11 + X X 1m + X 21 + X X 2m X nm The double summation operator can be thought of summing over a matrix [ ] Y = m X ij = m X ij = = 31 j=1 j=1 Lecture #4-9/05/2006 Slide 8 of 26

9 Bonus Coverage: The Product Operator Properties Double Bonus Coverage Just to be complete, there is also a product operator, usually denoted by a. This think just says multiply all elements inside of it... n X i = X 1 X 2 X 3 X n Example: X 1 = 3 X 2 = 5 X 3 = 4 n X i = = 60 Lecture #4-9/05/2006 Slide 9 of 26

10 Introduction Example Data Mode Median Mean Measure Behavior Now that we have put the summation operator to rest, we can now start talking about measures of central tendency or location. Three measures we will talk about will be the: Mean Median Mode These measures take our statistical distribution and condense it into a one-number summary. For each of these, we are trying to describe what the typical score looks like. Although there is no real definition of typical. Lecture #4-9/05/2006 Slide 10 of 26

11 Example Data Example Data Mode Median Mean Measure Behavior Recall last time we had some example data from the Gambling Research Instrument. To keep things consistent, we ll still use those data. The item we were using was #32: I become irritable when I am unable to gamble. 1. Never 2. Rarely 3. Occasionally 4. Sometimes 5. Often 6. Very Frequently Keep in mind we are assuming the integers we assign to the categories have equal intervals between them - recall the levels of measurement lecture. Lecture #4-9/05/2006 Slide 11 of 26

12 Example Data Behold, the data: Example Data Mode Median Mean Measure Behavior Frequency Histogram (Y axis has count of times response was observed) Lecture #4-9/05/2006 Slide 12 of 26

13 The Mode The mode is the most frequently occurring score. Example Data Mode Median Mean Measure Behavior To find the mode, simply find the maximum point on the graph. For our example, the mode is the score of 1. Technically speaking, for a categorical distribution (such as the one we have in our example) the mode is found by: Mode(X) = max{p(x = l)} l L For a continuous distribution, the mode is then just the point of the distribution function that has the highest Y-value (we will come to call this the likelihood) Mode(X) = max x X {f(x)} Lecture #4-9/05/2006 Slide 13 of 26

14 Pros and Cons of the Mode Cons: Example Data Mode Median Mean Measure Behavior Can have more than one mode. Very sensitive to the size and number of classes. Pros: Can be used on nominal and ordinal variables. Very robust to outliers. Lecture #4-9/05/2006 Slide 14 of 26

15 The Median Example Data Mode Median Mean Measure Behavior The median is the point that divides the distribution into two intervals that have an equal frequency. To find the median, you must first order the scores from lowest to highest. Then, determine which case (when N is odd) or pair of cases (when N is even) to take to find the Median. If N is odd, the median is the score of ordered observation number N+1 2. If N is even, the median is the average (midpoint) of ordered observation number N 2 and N Lets try this with the GRI data... Lecture #4-9/05/2006 Slide 15 of 26

16 The Median Example Data Mode Median Mean Measure Behavior Our sample has N = so we have to find the average of ordered case N 2 = 112/2 = 56 and N = 57 Here is our ordered data (20 numbers per row): Case 56, X (56) = 1...Case 57, X (57) = 1...so the median is 1. We could have found that from our bar chart: Lecture #4-9/05/2006 Slide 16 of 26

17 Pros and Cons of the Median Pros: Example Data Mode Median Mean Measure Behavior Can be used on nominal and ordinal variables. Is a very robust statistic (it will take a lot of observations to change the median drastically). Cons: As a member of order statistics, its theoretical statistical properties can be very difficult to derive. For instance, no one talks about the standard error of the median. Lecture #4-9/05/2006 Slide 17 of 26

18 The Mean Example Data Mode Median Mean Measure Behavior The Mean (also known as the average, expected value, arithmetic mean, or first moment) is a familiar measure of central tendency. The mean is found by taking the sum of all scores and dividing them by N: Mean = x = 1 N N The mean is the balance point of the distribution. To find the mean of our sample, we must know that x i N x i = 171 x = = Lecture #4-9/05/2006 Slide 18 of 26

19 Pros and Cons of the Mean Pros: Example Data Mode Median Mean Measure Behavior Is easy to compute. Is what we often think about when it comes to central tendency. Gives us easy to work with theoretical results. Cons: Not robust to outliers. When 64% of our sample had responded with a 1, the mean said the average response was Lecture #4-9/05/2006 Slide 19 of 26

20 Behavior of Three Measures Example Data Mode Median Mean Measure Behavior A nice way to summarize the three measure s sensitivity to outliers is to consider their behavior under differing distributions. I will draw along with this slide... Under symmetric distributions, Mean = Median = Mode. Under Positively Skewed distributions, Mean > Median > Mode. Under Negatively Skewed Distributions, Mean < Median < Mode. Lecture #4-9/05/2006 Slide 20 of 26

21 Example Properties of Example #2 As you saw in a previous slide, the mean is also called the expected value (or first moment) of a distribution. The name expected value comes from the statistical notion of expectation, another mathematical operator that will summarize a distribution. The notation for the expectation operator is E(X). For discrete distributions E(X) = x For continuous distributions, E(X) = xp(x). x xf(x)dx. In general, whatever goes within the () of E( ), multiplies the random variables density function in an expectation. For example E(2.323πX 3 ) x 2.323πX 3 p(x) Lecture #4-9/05/2006 Slide 21 of 26

22 Example To demonstrate the expectation, let s take a look at our GRI data, this time as a probability distribution. Example Properties of Example #2 Let s find E(X), which is E(X) = x Xp(X) X p(x) X p(x) As you see, E(X) = x... Lecture #4-9/05/2006 Slide 22 of 26

23 Properties of An important property of the Operator is: Example Properties of Example #2 E(a + cx) = a + ce(x) Special Cases for a and c as constants: E(a) = a E(cX) = ce(x) E(a + X) = a + E(X) With a discrete distribution (and using the properties of summation), can you show why these properties exist? Lecture #4-9/05/2006 Slide 23 of 26

24 Example #2 Let s find E(X 2 ), which is E(X 2 ) = x X 2 p(x) Example Properties of Example #2 X p(x) X 2 p(x) Lecture #4-9/05/2006 Slide 24 of 26

25 Final Thought Final Thought Next Class Today s lecture covered some basic descriptive measures for distributions. We talked about ways of summarizing the typical observation of a distribution. More importantly, we discussed expectation - something that will follow us throughout the course (and next semester, too). Next time we will show how we find one of our measures of variability (the variance) as an expectation. Lecture #4-9/05/2006 Slide 25 of 26

26 Next Time variability. Changes of metric (standardization, rescaling, etc...). More good clean fun with statistics. Final Thought Next Class Lecture #4-9/05/2006 Slide 26 of 26

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