Slide Number 1Slide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Some Common Examples of Parameters and StatisticsHealth Behavior Statistical Methods HP 340L Lecture 5 Describing Data: Measures of Variability Chapter 5Important Class Announcements • Home works are due at 12.00 noon of the day it is due • TA shall discuss the HW after a short lecture on Thursday • The first home work will be posted this Thursday • Late homework will be accepted in blackboard BUT will not be graded or accepted for the class • Syllabus has been revised to reflect the changes. We covered material in Kiess & Green Chapter 4  Descriptive statistics  Measures of central tendency • Mean, deviations from the mean, sums of squares • Median and mode • Effect of outliers Last Lecture We will cover material in Kiess & Green Chapter 5  Describing variability  Range, interquartile range (IQR)  Variance  Standard deviation Today’s Lecture The mean doesn’t tell the whole story  Example: two samples of 100 exam grades each; mean grade = 80 for both Descriptive Statistics Measures of:  Central tendency • Typical score  Dispersion • Amount of variability around the typical score • Range, Variance, Standard deviation Descriptive Statistics Indicate how much the scores differ, from each other and from the measure of central tendency  Another important characteristic of the data  Provides additional details to describe the scores  Critical component for many of the statistics to follow  Measures of variability include:  Range  Interquartile range (IQR)  Variance  Standard deviation Measures of Variability Range: The spread, or dispersion, in the scores  Basically the highest to the lowest score Range = xhighest - xlowest Range = 10 Range = 33 Measures of Variability Characteristics of the range  Range is determined by only 2 scores: the highest and the lowest  Can be inflated by outliers:  Outliers more likely with large samples  Unstable:  Change in only one score can greatly change the range Measures of VariabilityRange = 10 Range = 17 One additional score Measures of Variability Quartile Ranges  Interquartile range: x75 - x25  The range of scores of 50% of the subjects around the median  The range of the middle half of the scores Measures of Variability IQR = 4 Characteristics of the IQR  Not affected by outliers  Good for skewed distributions Measures of VariabilityInterquartile range: x75 - x25 = 63 – 59.25 = 3.75 Measures of Variability  Previous exam scores example  Use SPSS to generate a frequency table and descriptive statisticsMeasures of Variability  Previous exam scores example  Use SPSS to generate a frequency table and descriptive statistics Variance is the one of the most frequently used measure of variability  Is the average squared deviation from the mean Mean deviation Measures of Variability Population variance is a population parameter  Denoted by Greek letter, σ2  Measures how much the scores in the population vary around the population mean  Sums of squares divided by population size (N)  Unknown parameter, must be estimated by sample variance, s2 (statistic). ( )22 = XN−µσ∑Measures of Variability ( )22 = XN−µσ∑ Estimate the population variance from a sample (s2)  Step 1: Replace μ with sample estimate of mean and N with n (sample size):  Biased estimate, because it underestimates the population variance  Step 2: Correct for estimation of mean:  Bias can be corrected by using (n - 1) instead of n in the denominator Measures of Variability ( )( )2222= = XNxxsn−µσ−∑∑( )22= 1xxsn−−∑ Standard deviation (SD or s): most common measure of variability  Assesses how much the scores differ from the mean, on average  Expressed in the same units as the original measurement  Note: the units on the variance are squared  For many distributions, most scores are within one SD of the mean and nearly all are within two SDs of the mean Measures of Variability The sample SD is an estimate of the population SD  Expressed in the same units as the original measurement  Square root of the sample variance Measures of Variability ( )22= = 1xxssn−−∑ Interpreting the standard deviation  A typical distance of a data value from the mean  If a data distribution is approximately symmetric and bell-shaped, then” Measures of Variability  ±1 SD: encompasses 68.3% of the data  ±2 SD: encompasses 95.4% of the data  ±3 SD: encompasses 99.7% of the data Example: The number of emergency cases seen by six pediatrics residents in a 24-hour period 9, 8, 6, 4, 2, 1  What is the estimated variance? Measures of Variability ( )( )22= 15= 6152 = 10.45xxsnx−−−−=∑∑ Example: A recent driver’s education class took their driving exam and scored as follows: 98, 63, 92, 88, 92, 70, 98, 92, 75, 70 Measures of Variability  Calculate:  Range  IQR  Variance  Standard deviation Example: A recent driver’s education class took their driving exam and scored as follows: 98, 63, 92, 88, 92, 70, 98, 92, 75, 70 Range: IQR: Variance: Standard Deviation: Measures of Variability Example: A recent driver’s education class took their driving exam and scored as follows: 98, 63, 92, 88, 92, 70, 98, 92, 75, 70 Measures of Variability A researcher interested in the time demand of a statistics class on students  Five students were randomly selected to record the number of hours spent on homework per week: 13, 12, 6, 9, 10  Calculate range, variance, and standard deviation for this sample Measures of Variability A researcher interested in the time demand of a statistics class on students  Five students were randomly selected to record the number of hours spent on homework per week: 13, 12, 6, 9, 10 Measures of Variability A sample of 17-year-olds was surveyed about their use of alcohol  Asked how many times they had been drunk in