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

What you will learn in Chapter 2 Measures of central tendency Computing the mean and weighted mean for a set of scores Computing the mode using the mode and the median for a set of Selecting a measure of central tendency

Measures of Central Tendency  The AVERAGE is a single score that best represents a set of scores  Averages are also know as “Measure of Central Tendency”  Three different ways to describe the distribution of a set of scores… 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

Things to remember…  N = population n = sample  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  Average means the one measure that best represents a set of scores Different types of averages Type of average used depends on the question

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)

Weighted Mean Flying Proficiency Test(Salkind p. 23) ValueFrequencyValue*Freq , , , , , Total

You Try!! Using Weighted Mean to Find Average Super Bowl Yardage Penalty ValueFrequencyValue*Frequency 5 (ie. False starts, illegal downfield) 4 10 (offensive holding)4 11 (Half the distance penalties on kickoffs/punts) 3 15 (personal fouls)2 Total

Computing the Median  Median = point/score at which 50% of remaining scores fall above and 50% fall below.  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?

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

Cumm Percentage of Ages (N=20) Agesfreq%Cumm %

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

Using Calculator  Mode +. = statistical mode;  Shift +7= the mean “x-bar”  Shift +5= sum of x; square this value to get square of the sum;  Shift +4 = sum of squares  Shift +9= sample standard deviation  Shift+1=permutations  Shift+2=combinations  Shift+3= factorials

When to Use What…  Use the Mode when the data are categorical  Use the Median when you have extreme scores  Use the Mean when you have data that do not include extreme scores and are not categorical

Chapter 315 Measures of Central Tendency Choosing the right measure  Normal distribution Mean: = median/mode Median: = mean/mode Mode: = mean/median  They all work.  Pick the one that fits the need.

Chapter 316 Measures of Central Tendency Choosing the right measure  Positively skewed Mean: little high Median: middle score Mode: little low  Median works best

Chapter 317 Measures of Central Tendency Choosing the right measure  Negatively skewed Mean: too low Median: middle score Mode: little high  Median works best

Central Tendencies and Distribution Shape

Using SPSS

Glossary Terms to Know  Average  Measures of Central Tendency Mean  Weighted mean  Arithmetic mean Median  Percentile points  outliers Mode