Slide Number 1Slide Number 2Slide Number 3Slide Number 4Slide Number 5How to Describe DataSlide 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 30Slide Number 31Slide Number 32Slide Number 33Slide Number 34Slide Number 35Slide Number 36Health Behavior Statistical Methods HP 340L Lecture 4 Describing Data: Measures of Central Tendency Chapter 4 Homework 1 will be posted on blackboard next week  Due Thursday, February 2 Class Details We covered material in Kiess & Green Chapter 3  Graphing qualitative variables: bar and pie charts  Graphing quantitative variables: Histograms  Frequency distributions  Percentile ranks and percentiles Last Lecture We will cover material in Kiess & Green Chapter 4  Describing central tendency  Sample mean  Sample median  Sample mode Today’s Lecture Example: exam scores for 20 students 61 57 60 62 68 59 62 59 64 60 59 67 55 65 63 59 67 60 59 64  How do we make sense of these data?  What is the:  Most common score?  Lowest score?  Highest score? Graphing Quantitative (numerical) VariablesHP-340: Fall 2016 6 How to Describe Data • In terms of distribution. • In terms of central tendency. • In terms of spread or dispersion. Measures of:  Central tendency • Typical score  Dispersion • Amount of variability around the typical score • Range, Variance, Standard deviation Descriptive Statistics A frequency distribution is a tabular summary of the raw scores of a quantitative variable  What if we want to summarize raw scores by a single number?  Use measures of central tendency  Provides a sense of a ‘typical’ score  Mean: Arithmetic average  Median: Middle score  Mode: Most frequent score Measures of Central Tendency The most common measure of central tendency, denoted by a bar over the name of the variable, e.g.,  The sample mean for variable x, is the sum of the scores divided by the number of subjects in the sample (n): 1231... = nniixxx xxnxxnn=+ + ++==∑∑Sample Mean x Example: Prostate cancer is most common cancer (excluding skin melanoma) in men in the US  Prostate specific antigen (PSA) is a protein produced by cells of the prostate gland  Elevated PSA level in blood (>4.0 ng/ml) could indicate prostate cancer  Until recently, PSA screening was recommended for men age 50 or older, but too many false positives and false negatives Sample Mean PSA values for 20 healthy men age 40-49: 0.2, 0.2, 0.4, 0.5, 0.6, 0.6, 0.8, 0.8, 0.8, 0.9, 0.9, 1.1, 1.1, 1.1, 1.1, 1.2, 1.2, 1.5, 1.8, 4.2 Sample MeanSample Mean ( )0.2 0.2 0.4 0.5 0.6 ... 1.2 1.5 1.8 4.2= 20 1.05x+++++++++= Mean of the entire population is often denoted by the Greek letter μ Population Mean Can only be used for interval and ratio variables  Cannot be used with nominal or ordinal data  Uses all the scores  Any change of a score affects the mean  Very influenced by extreme scores (outliers):  One score may pull the mean away from the typical scores Characteristics of the Mean Effect of outlier on the mean  Data: 22, 34, 35, 35, 37, 38, 44 Mean = 35  Data: 22, 34, 35, 35, 37, 38, 144 Mean = 49.3 Characteristics of the Mean Outlier In PSA example:  PSA value of 4.2 might be an outlier  Mean with outlier = 1.05  Mean removing outlier = 0.884211 Characteristics of the MeanSum of all deviations from mean is 0 Mean is the “center of gravity” Characteristics of the Mean  Deviations from the mean: xD = (x – )  = 35 xxSubject x xD = (x- ) xD = (x- )2 1 22 -13 169 2 34 -1 1 3 35 0 0 4 35 0 0 5 37 2 4 6 38 3 9 7 44 9 81 Sum 245 0 264 xxSum of Squares (SS): Smaller than the squared deviation from any other number, e.g., median or mode The middle score, that has an equal number of scores above and below it  The 50th percentile  It cuts the distribution into two equal parts  Position of median:  The value of the median need not be represented in the original data Median 1Median = 2n+ To find the median:  First, arrange data (ordinal or interval) in order of size:  Example: The number of times 5 patients requested pain killers following a wisdom tooth extraction 1, 3, 4, 5, 9  What is the median?  For odd number of scores with no tied numbers around the median, the middlemost point in the distribution is the median: 4 Median Finding the median with even number of scores  The number of times 6 patients requested pain killers following a wisdom tooth extraction:  1, 3, 4, 5, 9, 11  What is the median?  For even number of scores with no tied numbers around the median, find the average of the two middle scores: Median ( )45Median = = 4.52+ The data represents the box scores for all the players on a basketball team and the points they scored : 2, 5, 6, 6, 8, 10, 12, 17  What is the median ? Median ()68 Median = = 72+Even number of scores, so: PSA example: 0.2, 0.2, 0.4, 0.5, 0.6, 0.6, 0.8, 0.8, 0.8, 0.9, 0.9, 1.1, 1.1, 1.1, 1.1, 1.2, 1.2, 1.5, 1.8, 4.2  What is the median PSA score? Median ( )0.9 0.9 Median = = 0.92+Even number of scores, so: The median is a percentile rank; specifically, the 50th percentile  The percentile rank of a score indicates the percentage of scores in the distribution that are equal to or less than that score  Cumulative frequency distributions are used to find percentile ranks MedianMedian Score Tally Frequency rf %f cf crf c%f 68 / 1 0.05 5 20 1.00 100 67 0 0 0 19 0.95 95 66 / 1 0.05 5 19 0.95 95 65 0 0 0 18 0.90 90 64 // 2 0.1 10 18 0.90 90 63 // 2 0.1 10 16 0.80 80 62 //// 4 0.2 20 14 0.70 70 61 /// 3 .0.15 15 10 0.50 50 60 // 2 0.1 10 7 0.35 35 59 / 1 0.05 5 5 0.25 25 58 // 2 0.1 10 4 0.20 20 57 / 1 0.05 5 2 0.10 10 56 0 0 0 1 0.05 5 55 / 1 0.05 5 1 0.05 5 For a score of 61, the percentile rank is 50 50% of the scores are equal to or less than 61  Example: our test score data from the previous lecture Effect of an outlier on the