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## Measures of Variability

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**Measures of Variability**Overheads 4**Quiz:Week 4 Understanding Check**The quiz is located under “Course Materials” You may want to stop the video and complete the quiz now. You are NOT required to submit this quiz.**Quiz: Question 1**1. What is the independent variable in Problem 1? • Type A and B Blood • Survival Rate • 80% chance of survival • Blood Type**Quiz: Question 2**2. What is the dependent variable in Problem 1? • Blood Type • Survival Rate • 80% chance of survival • Type A and B Blood**Quiz: Question 3**3. Which of the following would be most appropriately measured on an ordinal scale? • Type of Religion (Catholic, Protestant, Jewish…) • Gender (Male, Female) • Academic Ranking (Freshman, Sophomore, Junior, Senior) • Hair Color (Blonde, Brunette, Redhead…)**Quiz: Question 4**4. For Data 1, what is the mode of Group 1? • 3.00 • 4.00 • 5.00 • 6.00**Quiz: Question 5**4. For Data 1, what is the median of the entire data set? • 3.00 • 4.00 • 5.00 • 6.00**Quiz: Question 6**4. What is the mean of Group 2? • 3.40 • 4.60 • 4.00 • 5.00**Quiz: Question 7**7. What is the value of this symbol for Data 1? • 40.00 • 10.00 • 18.00 • 8.00**Measures of Variability**• How do we best describe the spread or dispersion of the scores in our distribution? • It depends on which measure of central tendency we consider. • Measures of Variability, some examples • For the Mode: Range • For the Median: Interquartile Range • For the Mean: Variance or Standard deviation**X**Figure 1**X**Figure 2**X**Figure 3**Mathematical Measures of Variability**• Nominal data: Range • X(max) – (X(min) • The best we can do to describe spread around a mode. • Notice, only two scores go into the calculation of the range. • Ordinal data: (Semi-) Interquartile range • Only takes the location of the scores (top half/bottom half) into account, not the value of the scores. Interquartile range Semi-interquartile range**Figure 4: Interquartile Range**25% 25% Q1 Q2 Q3**Mathematical Measures of Variability**• Nominal data: Range • X(max) – (X(min) • The best we can do to describe spread around a mode. • Notice, only two scores go into the calculation of the range. • Ordinal data: (Semi-) Interquartile range • Only takes the location of the scores (top half/bottom half) into account, not the value of the scores. Interquartile range Semi-interquartile range**Figure 5: Semi-Interquartile Range**25% 25% Q1 Q2 Q3**Q1 Q2 Q3**Normal Distribution Q3 – Q2 =Q2 – Q1**Negatively Skewed Distribution**Q3 – Q2 <Q2 – Q1 - + Q1 Q2 Q3**Positively Skewed Distribution**Q3 – Q2 >Q2 – Q1 - + Q1 Q2 Q3**Deviation Scores**• A deviation score is a measure of how far away from the mean each score falls. • The farther away a score is (meaning that it is in the tails of the distribution) the larger its deviation score. • If you added up all of the deviation scores they would equal zero. • This only happens when you calculate deviations around the mean.**The Mean**Sample Mean Population Mean**Deviation Scores**• A deviation score is a measure of how far away from the mean each score falls. • The farther away a score is (meaning that it is in the tails of the distribution) the larger its deviation score. • If you added up all of the deviation scores they would equal zero. • This only happens when you calculate deviations around the mean.**Deviation Scores**x1 = +5.00 x2 = - 2.00 X2 X1**Squared deviation scores.**• Since the deviation scores always add up to 0.00, then it doesn’t make sense to average them (since that would only give you 0.00/N). • So…instead we square the deviation scores before taking the average of the deviation scores. • This value is called the sums of squares.**Step 1**X x Raw Deviation Score Score**Step 2**X x x2 Raw Deviation Squared Score Score Deviation Score**Step 3**X x x2 Raw Deviation Squared Sum of Score Score Deviation Squared Score Deviation Scores**Formulas**2 Sum of the Sum of the Sum of the Squared Squared Scores Deviation Scores Scores Squared (Sum of Squares)**Least squares criterion.**• We already said that one property about the mean is that the sum of the deviation scores around the mean is equal to zero. • The other property of the mean is the least squares criterion. • If you square all of the deviation scores around the mean and add them together you will get a smaller value than if you did the same thing about any other number in a distribution (e.g., the median, the model).**Variance**• Variance of a population is the average of the squared deviations around the population mean (mu). Sum of Squared Deviations (Sums of Squares) Sample size**Calculations using the definitional formula for a**population. (σ2) • After you calculate the squared deviations and you add then together (the sums of squares) then you divide that value by your sample size.**Variance side note: Bias**• Variance for a population, σ2 • Variance for a sample, s2 : This adjustment is made because the sample produces a “biased” estimate of the population value**Calculations using the definitional formula for a sample**variance (s2). • After you calculate the squared deviations and you add then together (the sums of squares) then you divide that value by your sample size.**Standard Deviation**• Standard deviations are difficulty to interpret since they are usually utilized as a mathematically pleasing measure of variability. • Simply the square root of the variance. • They are useful for some inferences we will make later using the normal curve and probabilities. Population Standard Deviation Sample Standard Deviation**Definitional vs Computational**• Definitional Formula • Population • Definitional Formula • Sample • Computational Formula • Population • Computational Formula • Sample**Visual Displays and Variability**• We want graphs or plots that can show us how spread out our data are. • If scores are VERY spread out, then there is a lot of diversity amongst the individuals of our population (sample). • If this is true, then using a measure of central tendency**What does Variability look like?**• We use various graphical displays to examine the variability of the scores • Frequencies of continuous variables • Histogram • Stem-and-leaf • Grouped data • Boxplots (Box and whisker plots) • Bar charts, pie charts, etc.**In-Class Problem Set & Solution**Located under “Course Materials”**Writing a Results Section**For the online course you are not required to write a Results section