1 Describing distributions with numbers William P. Wattles Psychology 302

2 Measuring the Center of a distribution n Mean – The arithmetic average – Requires measurement data n Median – The middle value n Mode – The most common value

3 Measuring the center with the Mean

4 Our first formula

5 The Mean n One number that tells us about the middle using all the data. n The group not the individual has a mean.

Population Sample

6 Sample mean

7 Mu, the population mean

Population Sample

8 Calculate the mean with Excel n Save the file psy302 to your hard drive –right click on the file –save to desktop or temp n Open file psy302 n Move flower trivia score to new sheet

9 Calculate the mean with Excel n Rename Sheet – double click sheet tab, type flower n Calculate the sum – type label: total n Calculate the mean – type label: mean n Check with average function

10 Measuring the center with the Median n Rank order the values n If the number of observations is odd the median is the center observation n If the number of observations is even the median is the mean of the middle two observations. (half way between them)

11 Measuring the center with the Median

12 The mean versus the median n The Mean – uses all the data – has arithmetic properties n The Median – less influenced by Outliers and extreme values

Mean vs. Median

5 The Mean n The mean uses all the data. n The group not the individual has a mean. n We calculate the mean on Quantitative Data Three things to remember

n The mean tells us where the middle of the data lies. n We also need to know how spread out the data are.

Measuring Spread n Knowing about the middle only tells us part of the story of the data. n We need to know how spread out the data are.

Variability n Variety is the spice of life n Without variability things are just boring

Why is the mean alone not enough to describe a distribution? n Outliers is NOT the answer!!!!

The mean tells us the middle but not how spread out the scores are.

14 Example of Spread n New York n mean annual high temperature 62

14 Example of Spread n San Francisco n mean annual high temperature 65

16 Example of Spread n New York n meanmaxminrangesd n n San Francisco n

Example of Variability

17 Measuring Spread n Range n Quartiles n Five-number summary – Minimum – first quartile – median – third quartile – Maximum n Standard Deviation

n Mean 50.63% n Mean 33.19% Std Dev 21.4% Std Dev 13.2%

19 Deviation score n Each individual has a deviation score. It measures how far that individual deviates from the mean. n Deviation scores always sum to zero. n Deviation scores contain information. – How far and in which direction the individual lies from the mean

18 Measuring spread with the standard deviation n Measures spread by looking at how far the observations are from their mean. n The standard deviation is the square root of the variance. n The variance is also a measure of spread

Individual deviation scores

Standard deviation n One number that tells us about the spread using all the data. n The group not the individual has a standard deviation. Note !!

23 Standard Deviation

22 Variance

24 Properties of the standard deviation n s measures the spread about the mean n s=0 only when there is no spread. This happens when all the observations have the same value. n s is strongly influenced by extreme values

n New Column headed deviation n Deviation score = X – the mean

25 Calculate Standard Deviation with Excel n In new column type heading: dev2 n Enter formula to square deviation n Total squared deviations – type label: sum of squares n Divide sum of squares by n-1 – type label: variance

Moore page 50

n To Calculate Standard Deviation: n Total raw scores n divide by n to get mean n calculate deviation score for each subject (X minus the mean) n Square each deviation score n Sum the deviation scores to obtain sum of squares n Divide by n-1 to obtain variance n Take square root of variance to get standard deviation.

Population Sample

26 Sample variance

27 Population variance

Population Variance Sample Variance

28 Little sigma, the Population standard deviation

29 Sample standard deviation

Population Standard Deviation Sample Standard Deviation

To analyze data n 1. Make a frequency distribution and plot the data n Look for overall pattern and outliers or skewness n Create a numerical summary: mean and standard deviation.

41 Start with a list of scores

42 Make a frequency distribution

43 Frequency distribution

44 Represent with a chart (histogram)

45 Represent with line chart

Density Curve n Replaces the histogram when we have many observations.

Transform a score n Hotel Atlantico n 200 pesos n Peso a unit of measure

Transform a score n 1 dollar = pesos n 200/28.38=$7.05 n Dollar a unit of measure

31 n standardized observations or values. n To standardize is to transform a score into standard deviation units. n Frequently referred to as z-scores n A z-score tells how many standard deviations the score or observation falls from the mean and in which direction

32 Standard Scores (Z-scores) n individual scores expressed in terms of the mean and standard deviation of the sample or population. n Z = X minus the mean/standard deviation

33 Z-score

34 new symbols

35 Calculate Z-scores for trivia data n Label column E as Z-score n Type formula deviation score/std dev n Make std dev reference absolute (use F4 to insert dollar signs) n Copy formula down. n Check: should sum to zero

File extensions n Word.doc n Excel.xls n Text files.txt

To view File extensions n Open Windows Explorer n Choose Tools/Folder Options/View n uncheck “hide extensions for known file types.

37 Z Scores n Height of young women – Mean = 64 – Standard deviation = 2.7 n How tall in deviations is a woman 70 inches? n A woman 5 feet tall (60 inches) is how tall in standard deviations?

38 Z scores n Height of young women – Mean = 64 – Standard deviation = 2.7 n How tall in deviations is a woman 70 inches? z = 2.22 n A woman 5 feet tall (60 inches) is how tall in standard deviations? z = -1.48

39 Calculating Z scores

Calculating X from Z scores

72 Types of data n Categorical or Qualitative data –Nominal: Assign individuals to mutually exclusive categories. F exhaustive: everyone is in one category –Ordinal: Involves putting individuals in rank order. Categories are still mutually exclusive and exhaustive, but the order cannot be changed.

73 Types of data n Measurement or Quantitative Data –Interval data: There is a consistent interval or difference between the numbers. Zero point is arbitrary –Ratio data: Interval scale plus a meaningful zero. Zero means none. Weight, money and Celsius scales exemplify ratio data –Measurement data allows for arithmetic operations.

Review n Video2 Video2

60 The End

Mean vs. Median