CHAPTER 2 Percentages, Graphs & Central Tendency

Percentages

Graphing Step 1: Distribution -arrangement of scores in order of magnitude -RAW DATA of stress ratings of 30 students: 8,7,4,10,8,6,8,9,9,7,3,7,6,5,0,9,10,7,7,3,6,7,5,2,1,6,7,10,8,8 -Distributed scores: 0,1,2,3,3,4,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,8,9,9,9,10,10,10 Step 2: Frequency Distribution -arrangement of scores in order of magnitude with the number of times each score occurred

Graphing Step 3: Make a graph -raw scores go on the horizontal line (X-axis or abscissa) -frequency goes on the vertical line (Y-axis or ordinate) Step 4: Make a histogram -rectangular bar drawn above each raw score **bar graphs are used for noncontinuous or categorical data (EX: college major) -steps between values are separate **histograms are used for continuous data (EX: SAT scores) -steps between values are not separate Step 5: Make a frequency polygon -single point used to show the frequency of each score & points are connected with lines -especially useful when showing two distributions simultaneously

Graphing Stem & Leaf Graph -just another way to graph data -first digit in stem column (EX: 9 represents scores in the 90’s) -trailing digit in leaf column (EX: 0 represents 90 & 4 represents 94) WOW! Graph -A deceptive graph due to not setting the base of the ordinate at 0 -Some corporations hire statisticians to create wow graphs so their annual sales report looks better!

Normal and kurtotic distributions -Normal curve (mesokurtic): bell-shaped & symmetrical *standard of comparison because distributions observed in nature usually take this shape -Kurtosis: extent to which a frequency distribution deviates from the normal curve *Leptokurtic: high & peaked --more scores in the tails than the normal curve *Platykurtic: low & flat --less scores in the tails than the normal curve Shapes of Frequency Distributions Carl Gauss “Father of the Normal Curve” MesokurticLeptokurticPlatykurtic

Shapes of Frequency Distributions Non-bell shaped -Unimodal distribution: one value clearly occurs more than any other -Bimodal distribution: two values clearly occur more than any other -Rectangular distribution: all values occur equally

Shapes of Frequency Distributions Symmetrical and Non-symmetrical distributions -usually distributions look “normal” or symmetrical -skewed distribution: a distribution that is not symmetrical *the side with fewer scores is considered the direction of the skew --positively skewed: skewed to the right --negatively skewed: skewed to the left *usually happens when what is being measured has an upper or lower limit --EX: how many kids do each of you have? --Can’t have less than 0 so the distribution will probably look positively skewed Symmetrical Non- Symmetrical/Skewed to the right (positively skewed) Non- Symmetrical/Skewed to the left (negatively skewed)

Question: How can a group of scores be summarized with a single number? Answer: Central Tendency! -The typical or most representative value of a group of scores Includes the mean, median & mode Central Tendency

Central Tendency: Mean Sum of all the scores divided by the number of scores Formula for the mean: Σ (sigma) means “summation of” X stands for raw score N stands for entire number of observations M stands for mean  Most stable measure of central tendency because all the scores in a distribution are included in its calculation (not true to mode or median)  Use with equal-interval variables: equal amount between numbers (EX: age, weight, GPA)

Central Tendency: Median Median (Mdn): middle score when all the scores in a distribution are arranged from highest to lowest -If you have even numbers, calculate the mean between the 2 middle numbers Better to use than the mean when there are extreme scores (outliers) -EX: Calculate the mean and median for these scores:.74,.86, 2.32,.79,.81 -Which represents the data better? Use with rank-ordered variables: numeric variable in which values are ranks (EX: 1 st place)

Central Tendency:Mode Mode (Mo): value with the greatest frequency in a distribution -aka: the highest point on a histogram or frequency polygon In a perfectly symmetrical bimodal distribution the mode & mean are the same -if the distribution doesn’t look this way then the mode is usually unrepresentative Rarely used in psychology research Used with nominal variables: values that are categories (EX: gender)

Central Tendency: Skewness If the distribution has little or no skew then the median, mode & mean should be the same or close In skewed distributions (due to extreme scores), the mean is “pulled” toward the tail of the distribution & is unrepresentative of the body of scores. In a negatively skewed distribution, the mean is lower than the median (a) In a positively skewed distribution, the mean is higher than the median (b)