Psych 230 Psychological Measurement and Statistics. Pedro Wolf September 9, 2009. So Far. Stem and leaf plots Bar plots Summarizing scores using Frequency how a frequency distribution is created Graphing frequency distributions bar graphs, histograms, polygons Types of distribution

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Psych 230 Psychological Measurement and Statistics

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So Far • Stem and leaf plots • Bar plots • Summarizing scores using Frequency • how a frequency distribution is created • Graphing frequency distributions • bar graphs, histograms, polygons • Types of distribution • normal, skewed, bimodal • Relative frequency and the normal curve

the Normal Curve How likely is it that a certain score will occur?

Today…. • Summarizing scores using central tendency • what is central tendency? • The Mode • what it is, how to calculate it, & when to use it • The Median • what it is, how to calculate it, & when to use it • The Mean • what it is, how to calculate it, & when to use it • applying the mean to research

Range • what it is & how to calculate it • Variance • what it is & how to calculate it • Standard Deviation • what it is & how to calculate it • Variability and the Normal Distribution • Population Variance and Standard Deviation

Why do we need a measure of Central Tendency? • Often we would like to know the most typical or representative score of a dataset • How many drinks do students consume a week? • What are the political beliefs of students? • What is people’s favorite color? • How much do lawyers get paid? • What is the temperature in London? • There are different ways to calculate a typical score. • Each way has advantages and disadvantages. Depends on: • Type of data • Distribution of data

What is a Measure of Central Tendency? • Measures of central tendency answer the question: • “Are the scores generally high scores or generally low scores?” • Allow us to compare values: • Average high / low temp in May in Tucson: 90ºF / 53ºF • Average high / low temp in May in St. Petersburg: 59ºF / 42ºF • A statistic that indicates where the center of the distribution tends to be located

Measures of Central Tendency • There are three commonly used measures of central tendency • Mode • Median • Mean • There is no single, perfect, measure of central tendency

Example • The following are the salaries of the 15 employees of a small consulting company $82,000 $64,000 $36,400 $34,000 $29,200 $29,200 $29,200 $28,000 $26,800 $26,800 $26,800 $24,400 $24,400 $24,400 $24,400 • What is the typical salary of an employee in this company? • How can different measures of central tendency be used to make different arguments?

What is the Mode? • The mode is the score that has the highest frequency in the data • The mode is always used to describe central tendency when the scores reflect a nominal scale of measurement • Can also be used for other scales of measurement • Scores: 2,3,4,4,5,5,5 • Mode=5

How to find the Mode • Can find the mode by inspection (as opposed to computation) • Simply the score with the highest frequency

Example - Mode from Raw Scores • What is the mode of the following data: 14 10 1 2 15 4 14 14 13 15 11 15 13 2 12 13 14 13 14 25 47 1 14 15

Example - Mode from Raw Scores • What is the mode of the following data: 14 10 1 2 15 4 1414 13 15 11 15 13 2 12 13 14 13 14 25 47 1 14 15 Mode = 14

Unimodal Distributions • When a graph has one hump (such as on the normal curve) the distribution is called unimodal

Bimodal Distributions • When a graph shows two scores that are tied for the most frequently occurring score, it is called bimodal.

Example • The following are the salaries of the 15 employees of a small consulting company. $82,000 $64,000 $36,400 $34,000 $29,200 $29,200 $29,200 $28,000 $26,800 $26,800 $26,800 $24,400 $24,400 $24,400 $24,400 • What is the modal salary? • $24,400 • is this a good description of the typical salary?

Mode • Advantages: • can be used with nominal data • easily identified • unaffected by extreme scores • bimodal datasets may suggest interesting subgroups • Disadvantages • not necessarily a unique score • not very precise • cannot be manipulated mathematically

What is the Median? • The median is the middle score of the data; the score that divides the data in half • The median is the score at the 50th percentile • you did this in your homework when calculating the quartiles • The median is used to summarize ordinal or highly skewed interval or ratio scores

How to Find the Median • When data are normally distributed, the median is the same score as the mode. • When data are not normally distributed, follow the following procedure: • arrange the scores from lowest to highest. • if there are an odd number of scores, the median is the score in the middle position. • if there are an even number of scores, the median is the average of the two scores in the middle. • Median score = (N+1)/2

Example - Median from Raw Scores • What is the median of the following data: 14 10 1 2 15 4 14 14 13 15 11 15 13 2 12 13 14 13 14 25 47 1 14 15

Example - Median from Raw Scores • What is the median of the following data: 14 10 1 2 15 4 14 14 13 15 11 15 13 2 12 13 14 13 14 25 47 1 14 15 • First, arrange in order of magnitude 1 1 2 2 4 10 11 12 13 13 13 13 14 14 14 14 14 14 15 15 15 15 25 47

Example - Median from Raw Scores 1 1 2 2 4 10 11 12 13 13 13 13 14 14 14 14 14 14 15 15 15 15 25 47 • Number of scores (N) = • N=24 • Median is the average of the middle two: • (N+1)/2 = (24+1)/2 = 25/2 = 12.5 • Average of the 12th and 13th score • (13 + 14) / 2 = 13.5

Example - Median from Frequency Table • What is the median of the following data: Xf N=17 6 5 5 6 Median = (N+1)/2 = (17+1)/2 = 4 3 9th score 3 1 2 1 1 1

Example - Median from Frequency Table • What is the median of the following data: Xf N=17 6 5 5 6 Median = (N+1)/2 = (17+1)/2 = 4 3 9th score 3 1 2 1 Median = 5 1 1

Median • Advantages: • useful for skewed distributions • unaffected by extreme scores • useful for dividing sets of scores in to two halves (for example, high and low scorers in an exam) • Disadvantages • does not take into account extreme scores • cannot be manipulated mathematically

What is the Mean? • The mean is the score located at the exact mathematical center of a distribution • the “average” • The mean is used to summarize interval or ratio data in situations when the distribution is symmetrical and unimodal • By far the most commonly used measure of central tendency

How to Find the Mean • The symbol for the sample mean is • The formula for the sample mean is:

Example • Calculate the mean of the following data: • 12, 15, 17, 12, 13, 9, 1, 6, 3, 12, 12, 16, 17 • Mode = 12 • Median = 12 N = 13 S X = 12+15+17+12+13+9+1+6+3+12+12+16+17 S X = 145

Example • Calculate the mean of the following data: • 12, 15, 17, 12, 13, 9, 1, 6, 3, 12, 12, 16, 17 • Mode = 12 • Median = 12 N = 13 S X = 12+15+17+12+13+9+1+6+3+12+12+16+17 S X = 145 = S X / N = 145 / 13 = 11.154

Example • The following are the salaries of the 15 employees of a small consulting company $82,000 $64,000 $36,400 $34,000 $29,200 $29,200 $29,200 $28,000 $26,800 $26,800 $26,800 $24,400 $24,400 $24,400 $24,400 • What is the mean salary? • 510,000/15 = $34,000

Sample Mean vs. Population Mean • is the sample mean. This is a sample statistic. • The mean of a population is a parameter. It is symbolized by (pronounced “mew”) • is used to estimate the corresponding population mean

Your Turn - Mean • For the mean, we need SX and N • We know that N = 18 • What is SX?

Your Turn - Mean • For the mean, we need SX and N • We know that N = 18 • What is SX? • 10+11+12+13+13+13+13+14+14+14+14+14+14+15+15+15+15+17 SX=246

Weighted mean • The mean of a group of means • Sometimes you want to compare groups with different numbers of scores • Suppose you have 4 class averages: 75, 78, 72, 80. • How do you find the mean? • (75+78+72+80)/4 = 76.25 • Only works if every class has the same number of people

Central Tendency - Normal Distributions • On a perfect normal distribution, all three measures of central tendency are located at the same score: mean=median=mode