Chapter 2 Means to an End: Computing and Understanding Averages Part II  igma Freud & Descriptive Statistics

Measures of Central Tendency What is Central Tendency? Three different measures of central tendency… or “averages” Mean – typical average score Median – middle score Mode – most common score What is Central Tendency? Three different measures of central tendency… or “averages” Mean – typical average score Median – middle score Mode – most common score

Computing the Mean Formula for computing the mean “X bar” is the mean value of the group of scores “  ” (sigma) tells you to add together whatever follows it X is each individual score in the group The n is the sample size Formula for computing the mean “X bar” is the mean value of the group of scores “  ” (sigma) tells you to add together whatever follows it X is each individual score in the group The n is the sample size

Things to remember… N = population size n = sample size Sample mean is the measure of central tendency that best represents the population mean Mean is VERY sensitive to extreme scores that can “skew” or distort findings – called “Outliers” “Average” could refer to mean, median or mode… must specify. N = population size n = sample size Sample mean is the measure of central tendency that best represents the population mean Mean is VERY sensitive to extreme scores that can “skew” or distort findings – called “Outliers” “Average” could refer to mean, median or mode… must specify.

Example: Car Mileage Case Sample mean for five car mileages 30.8, 31.7, 30.1, 31.6, 32.1 LO1 3-5

Computing the Median Median = point/score at which half of the remaining scores fall below and half fall above. NO standard formula Rank order scores from highest to lowest or lowest to highest Find the “middle” score BUT… What if there are two middle scores? What if the two middle scores are the same? Median = point/score at which half of the remaining scores fall below and half fall above. NO standard formula Rank order scores from highest to lowest or lowest to highest Find the “middle” score BUT… What if there are two middle scores? What if the two middle scores are the same?

Example: Car Mileage Case Example 3.1: First five observations from Table 3.1: 30.8, 31.7, 30.1, 31.6, 32.1 In order: 30.1, 30.8, 31.6, 31.7, 32.1 There is an odd so median is one in middle, or 31.6 LO1 3-7

Weighted Mean Example List all values for which the mean is being calculated (list them only once) List the frequency (number of times) that value appears Multiply the value by the frequency Sum all Value x Frequency Divide by the total Frequency (total n size) List all values for which the mean is being calculated (list them only once) List the frequency (number of times) that value appears Multiply the value by the frequency Sum all Value x Frequency Divide by the total Frequency (total n size)

A little about Percentiles… Percentile points are used to define the percent of cases equal to and below a certain point on a distribution (i.e. data set). 75 th %tile – means that the score received is at or above 75 % of all other scores in the distribution 25 th %tile – means that the score received is at or above 25 % of all other scores in the distribution “Norm-referenced” measure allows you to make comparisons Percentile points are used to define the percent of cases equal to and below a certain point on a distribution (i.e. data set). 75 th %tile – means that the score received is at or above 75 % of all other scores in the distribution 25 th %tile – means that the score received is at or above 25 % of all other scores in the distribution “Norm-referenced” measure allows you to make comparisons

Percentiles and Quartiles For a set of measurements arranged in increasing order, the p th percentile is a value such that p percent of the measurements fall at or below the value and (100-p) percent of the measurements fall at or above the value The first quartile Q 1 is the 25th percentile The second quartile Q 2 (median) is the 50 th percentile The third quartile Q 3 is the 75th percentile The interquartile range IQR is Q 3 - Q

Computing the Mode Mode = most frequently occurring score NO formula List all values in the distribution Tally the number of times each value occurs The value occurring the most is the mode Democrats = 90 Republicans = 70 Independents = 140: the MODE!! When two values occur the same number of times -- Bimodal distribution Mode = most frequently occurring score NO formula List all values in the distribution Tally the number of times each value occurs The value occurring the most is the mode Democrats = 90 Republicans = 70 Independents = 140: the MODE!! When two values occur the same number of times -- Bimodal distribution

When to Use What… Use the Mode when the data are categorical (example: # of males vs. females) Use the Median when you have extreme scores (outliers) Use the Mean when you have data that do not include extreme scores and are not categorical Use the Mode when the data are categorical (example: # of males vs. females) Use the Median when you have extreme scores (outliers) Use the Mean when you have data that do not include extreme scores and are not categorical

Using SPSS