Tuesday August 27, 2013 Distributions: Measures of Central Tendency & Variability

Today: Finish up Frequency & Distributions, then Turn to Means and Standard Deviations First, hand in your homework. Any questions from last time?

Grouped Frequency Table A frequency table that uses intervals (range of values) instead of single values Pairs of Shoes X ValuesFreq %Cumulative %↑Cumulative %↓ Total24100

Frequency Graphs  Histogram  Plot the different values against the frequency of each value

Frequency Graphs  Histogram (create one for class height)  Step 1: make a frequency distribution table (may use grouped frequency tables)  Step 2: put the values along the bottom, left to right, lowest to highest  Step 3: make a scale of frequencies along left edge  Step 4: make a bar above each value with a height for the frequency of that value

Frequency Graphs  Frequency polygon - essentially the same, but uses lines instead of bars

Properties of distributions Distributions are typically summarized with three features  Shape  Center  Variability (Spread)

Shapes of Frequency Distributions  Unimodal, bimodal, and rectangular

Shapes of Frequency Distributions  Symmetrical and skewed distributions  Normal and kurtotic distributions

Next Topic In addition to using tables and graphs to describe distributions, we also can provide numerical summaries

Chapters 3 & 4 Measures of Central Tendency ◦ Mean ◦ Median ◦ Mode Measures of Variability ◦ Standard Deviation & Variance (Population) ◦ Standard Deviation & Variance (Samples) Effects of linear transformations on mean and standard deviation

Self-Monitor you Understanding These topics should all be review from PSY 138, so I will move fairly quickly through the lecture. I will stop periodically to ask for questions. Please ask if you don’t understand something!!! If you are confused by this material, it will be very hard for you to follow and keep up with later topics.

Describing distributions Distributions are typically described with three properties: ◦ Shape: unimodal, symmetrical, skewed, etc. ◦ Center: mean, median, mode ◦ Spread (variability): standard deviation, variance

Describing distributions Distributions are typically described with three properties: ◦ Shape: unimodal, symmetric, skewed, etc. ◦ Center: mean, median, mode ◦ Spread (variability): standard deviation, variance

Which center when? Depends on a number of factors, like scale of measurement and shape. ◦ The mean is the most preferred measure and it is closely related to measures of variability ◦ However, there are times when the mean isn’t the appropriate measure.

Which center when? Use the median if:  The distribution is skewed  The distribution is ‘open-ended’  (e.g. your top answer on your questionnaire is ‘5 or more’)  Data are on an ordinal scale (rankings) Use the mode if the data are on a nominal scale

Self-monitor your understanding We are about to turn to a discussion of calculating means. Before we move on, any questions about when to use which measure of central tendency?

The Mean The most commonly used measure of center The arithmetic average ◦ Computing the mean – The formula for the population mean is (a parameter): – The formula for the sample mean is (a statistic): Add up all of the X’s Divide by the total number in the population Divide by the total number in the sample Note: Sometimes ‘ ’ is used in place of M to denote the mean in formulas

The Mean Number of shoes: 2,2,2,5,5,5,7,8 6,10,10,12,12,13,14,14,15,15,20,20,20,20,25,30 Suppose we want the mean of the entire group? Can we simply add the two means together and divide by 2? ( )/2 = 20.5/2 = NO. Why not? = ( )/8 = 36/8 = 4.5 = ( )/16 = 256/16 = 16

The Weighted Mean Number of shoes: 2,2,2,5,5,5,7,8,6,10,10,12,12,13,14,14,15,15,20,20,20,20,25,30 Mean for men = 4.5Mean for women = 16 = [(4.5*8)+(16*16)]/(8+16) =(36+256)/24) = 292/24 = Need to take into account the number of scores in each mean ( & )

The Weighted Mean Number of shoes: 2,2,2,5,5,5,7,8,6,10,10,12,12,13,14,14,15,15,20,20,20,20,25,30 Both ways give the same answer Let’s check: = [(4.5*8)+(16*16)]/(8+16) = (36+256)/24 = 292/24 = = 256/24=12.17

Self-monitor your understanding We are about to move on to a quick discussion of calculating the median and mode. Before we move on, any questions about the formulae for the population mean, sample mean? Questions about the weighted mean?

The median The median is the score that divides a distribution exactly in half. Exactly 50% of the individuals in a distribution have scores at or below the median. ◦ Case1: Odd number of scores in the distribution Step1: put the scores in order Step2: find the middle score Step1: put the scores in order Step2: find the middle two scores Step3: find the arithmetic average of the two middle scores –Case2: Even number of scores in the distribution

The mode The mode is the score or category that has the greatest frequency. ◦ So look at your frequency table or graph and pick the variable that has the highest frequency. so the mode is 5 so the modes are 2 and 8 Note: if one were bigger than the other it would be called the major mode and the other would be the minor mode major mode minor mode

Self-monitor your understanding We are about to switch to the topic of measures of variability Before we move on, any questions about measures of central tendency?

Describing distributions Distributions are typically described with three properties: ◦ Shape: unimodal, symmetric, skewed, etc. ◦ Center: mean, median, mode ◦ Spread (variability): standard deviation, variance

Variability of a distribution Variability provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together. ◦ In other words variabilility refers to the degree of “differentness” of the scores in the distribution. High variability means that the scores differ by a lot Low variability means that the scores are all similar

Standard deviation The standard deviation is the most commonly used measure of variability. ◦ The standard deviation measures how far off all of the scores in the distribution are from the mean of the distribution. ◦ Essentially, the average of the deviations. μ

Computing standard deviation (population) Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution. Our population 2, 4, 6, = -3 μ X - μ = deviation scores -3

Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution. Our population 2, 4, 6, = = -1 μ X - μ = deviation scores Computing standard deviation (population)

Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution. Our population 2, 4, 6, = = = +1 μ X - μ = deviation scores 1 Computing standard deviation (population)

Step 1: Compute the deviation scores: Subtract the population mean from every score in the distribution. Our population 2, 4, 6, = = = = +3 μ X - μ = deviation scores 3 Notice that if you add up all of the deviations they must equal 0. Computing standard deviation (population)

Step 2: Get rid of the negative signs. Square the deviations and add them together to compute the sum of the squared deviations (SS). SS = Σ (X - μ) = = = = +3 X - μ = deviation scores = (-3) 2 + (-1) 2 + (+1) 2 + (+3) 2 = = 20 Computing standard deviation (population)

Step 3: Compute the Variance (the average of the squared deviations)  Divide by the number of individuals in the population. variance = σ 2 = SS/N Computing standard deviation (population)

Step 4: Compute the standard deviation. Take the square root of the population variance. standard deviation = σ = Computing standard deviation (population)

To review: ◦ Step 1: compute deviation scores ◦ Step 2: compute the SS  SS = Σ (X - μ ) 2 ◦ Step 3: determine the variance  take the average of the squared deviations  divide the SS by the N ◦ Step 4: determine the standard deviation  take the square root of the variance Computing standard deviation (population)

Self-monitor your understanding We are about to learn how to calculate sample standard deviations. Before we move on, any questions about how to calculate population standard deviations? Any questions about these terms: deviation scores, squared deviations, sum of squares, variance, standard deviation? Any questions about these symbols : SS

The basic procedure is the same. ◦ Step 1: compute deviation scores ◦ Step 2: compute the SS ◦ Step 3: determine the variance  This step is different ◦ Step 4: determine the standard deviation Computing standard deviation (sample)

Step 1: Compute the deviation scores ◦ subtract the sample mean from every individual in our distribution. Our sample 2, 4, 6, = = = = +3 X - M = Deviation Score M

Step 2: Determine the sum of the squared deviations (SS). Computing standard deviation (sample) = = = = +3 = (-3) 2 + (-1) 2 + (+1) 2 + (+3) 2 = = 20 X - M = deviation scores SS = Σ(X - M) 2 Apart from notational differences the procedure is the same as before

Step 3: Determine the variance Computing standard deviation (sample) Population variance = σ 2 = SS/N Recall: μ X 1 X 2 X 3 X 4 The variability of the samples is typically smaller than the population’s variability

Step 3: Determine the variance Computing standard deviation (sample) Population variance = σ 2 = SS/N Recall: The variability of the samples is typically smaller than the population’s variability Sample variance = s 2 To correct for this we divide by (n-1) instead of just n

Step 4: Determine the standard deviation standard deviation = s = Computing standard deviation (sample)

Self-monitor your understanding Next, we’ll find out how changing our scores (adding, subtracting, multiplying, dividing) affects the mean and standard deviation. Before we move on, any questions about the sample standard deviation? About why we divide by (n-1)? About the following symbols: ◦s2◦s2 ◦s◦s

Properties of means and standard deviations Change/add/delete a given score MeanStandard deviation changes –Changes the total and the number of scores, this will change the mean and the standard deviation

Properties of means and standard deviations –All of the scores change by the same constant. M old Change/add/delete a given score MeanStandard deviation Add/subtract a constant to each score changes

Properties of means and standard deviations –All of the scores change by the same constant. Change/add/delete a given score MeanStandard deviation changes Add/subtract a constant to each score M old

Properties of means and standard deviations –All of the scores change by the same constant. Change/add/delete a given score MeanStandard deviation changes Add/subtract a constant to each score M old

Properties of means and standard deviations –All of the scores change by the same constant. Change/add/delete a given score MeanStandard deviation changes Add/subtract a constant to each score M old

Properties of means and standard deviations –All of the scores change by the same constant. –But so does the mean Change/add/delete a given score MeanStandard deviation changes Add/subtract a constant to each score changes M new

Properties of means and standard deviations –It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Change/add/delete a given score MeanStandard deviation changes Add/subtract a constant to each score changes M old

Properties of means and standard deviations –It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Change/add/delete a given score MeanStandard deviation changes Add/subtract a constant to each score changes M old

Properties of means and standard deviations –It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Change/add/delete a given score MeanStandard deviation changes Add/subtract a constant to each score changes M old

Properties of means and standard deviations –It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Change/add/delete a given score MeanStandard deviation changes Add/subtract a constant to each score changes M old

Properties of means and standard deviations –It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Change/add/delete a given score MeanStandard deviation changes Add/subtract a constant to each score changes M old

Properties of means and standard deviations –It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Change/add/delete a given score MeanStandard deviation changes Add/subtract a constant to each score changes M old

Properties of means and standard deviations –It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Change/add/delete a given score MeanStandard deviation changes Add/subtract a constant to each score changes M old

Properties of means and standard deviations –It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Change/add/delete a given score MeanStandard deviation changes No changechangesAdd/subtract a constant to each score M old M new

Properties of means and standard deviations Change/add/delete a given score MeanStandard deviation Multiply/divide a constant to each score changes No changechangesAdd/subtract a constant to each score = = +1 (-1) 2 (+1) 2 s = M

Properties of means and standard deviations –Multiply scores by 2 Change/add/delete a given score MeanStandard deviation Multiply/divide a constant to each score changes No changechanges Add/subtract a constant to each score = = +2 (-2) 2 (+2) 2 s = M S old =1.41