Practice Page 65 –2.1

Positive Skew

Note Slides online

Histogram

Frequency Polygon

Stem-and-Leaf Display Stem-and-leaf display with a bigger data set Note: The stem-and-leaf is like a histogram turned sideways!

Describing Distributions Bell-shaped distribution

Describing Distributions

Kurtosis The relative concentration of scores in the center of the distribution Mesokurtic

Kurtosis The relative concentration of scores in the center of the distribution Platykurtic

Kurtosis The relative concentration of scores in the center of the distribution Leptokurtic

Measures of Central Tendency Give one value that represents an entire group of scores Mean Median Mode

Mean On your tests you get: 70%, 80%, 80%, 90% The mean is 80% You know how to do this!

Mean = the mean = an instruction to add (sigma) “the sum of” = a score = the number of scores

Practice What is the mean of: 5, 8, 6, 3, 2, 2, 9 Mean = 35 / 7 = , 11.6, 12.9, 14.7, 10.5, 11.9, 20.2, 15.5 Mean = / 8 = 13.48

Characteristics of the mean If the mean is subtracted from each score and the differences added, the sum will equal zero

Characteristics of the mean 70, 80, 80, 90 Mean = = = = 10  = 0

Characteristics of the mean The mean is the point about which the sum of the squared deviations is minimized

Characteristics of the mean 70, 80, 80, 90 Mean = = = = 0 2 = = 10 2 =100  =200

Population vs. Sample _ x = The mean of a sample  = The mean of a population *They are both calculated the same way!

Population vs. Sample

The Median The point that divides a distribution of scores into two parts that are equal in size

The Median 10, 5, 13, 6, 14, 17, 2, 6, 9 2, 5, 6, 6, 9, 10, 13, 14, 17

The Median 10, 5, 13, 6, 14, 17, 2, 6, 9 2, 5, 6, 6, 9 10, 13, 14, 17

The Median 5, 8, 9, 15, 20, 25, 50

The Median 5, 8, 9, 15 20, 25, 50

The Median 5, 8, 9, 12, 15, 18, 22, 30, 32, 40

The Median 5, 8, 9, 12, 15, 18, 22, 30, 32, 40

The Median 5, 8, 9, 12,15, 18,22, 30, 32,

Practice What is the median of: 5, 8, 6, 3, 2, 2, 9 2, 2, 3, 5, 6, 8, 9 (7+1) / 2 = 4 Median = 5

Practice What is the median of: 10.5, 11.6, 12.9, 14.7, 10.5, 11.9, 20.2, , 10.5, 11.6, 11.9, 12.9, 14.7, 15.5, 20.2 (8+1) / 2 = 4.5 ( ) / 2 = 12.4 Median = 12.4

The Mode The most frequently occurring score 5, 6, 8, 9, 10, 10, 10, 12, 14, 14 Mode = 10

Practice What is the mode of: 5, 8, 6, 3, 2, 2, 9 Mode = , 11.6, 12.9, 14.7, 10.5, 11.9, 20.2, 15.5 Mode = 10.5

Determining Skewness

For example: Mean = 4 Median = 10 Mean Median

Determining Skewness For example: Mean = 10 Median = 4 Median Mean

Determining Skewness Mean < Median = Negative Skew Mean > Median = Positive Skew Mean = Median = No Skew

Which should you use?

What is the mean, median, and mode?

Mean = 492 / 38 = 12.95

Mode = 14

(38 + 1)/ 2 = 19.5 Median = 14

The Test Scores of 3 Students Joe = 78, 60, 92, 80, 80 Bob = 47, 100, 98, 45, 100 Mary = 78, 79, 77, 78, 78

The Test Scores of 3 Students Joe = 78, 60, 92, 80, 80 –Mean = 78 Bob = 47, 100, 98, 45, 100 –Mean = 78 Mary = 78, 79, 77, 78, 78 –Mean = 78

Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together

Range The highest score minus the lowest score Joe = 78, 60, 92, 80, 80 Range = = 32

Range The highest score minus the lowest score Bob = 47, 100, 98, 45, 100 Range = = 55

Range The highest score minus the lowest score Mary = 78, 79, 77, 78, 78 Range = = 2

The Test Scores of 3 Students Joe = 78, 60, 92, 80, 80 –Mean = 78 Range = 32 – Bob = 47, 100, 98, 45, 100 –Mean = 78Range = 55 Mary = 78, 79, 77, 78, 78 –Mean = 78Range = 2

Range In general - the larger the range score, the more variance Pro: Easy to calculate Con: The range only depends on two extreme scores; can be misleading

Range 20, 62, 54, 32, 28, 44, 72, 69, 50 1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,5, 5, 5, 5, 5, 99

Range 20, 62, 54, 32, 28, 44, 72, 69, 50 Range = 49 1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,5, 5, 5, 5, 5, 99 Range = 98!!

Interquartile Range The range of scores that make up the middle 50 percent of the distribution Need to find the 25th percentile score and the 75th percentile score

Interquartile Range 50%

Interquartile Range. 25 (N) = The location of the 25th percentile score counting from the bottom.25 (N) = The location of the 75th percentile score counting from the top N = the number of cases *If the answer is not even simply average *Similar to how you found the median!!

Interquartile Range IQR = 75th percentile - 25th percentile

Interquartile Range 2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99

Interquartile Range 6 2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, (12) = 3 Counting 3 from the bottom the 25th percentile score = 6

Interquartile Range 62, 2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, (12) = 3 Counting 3 from the top the 75th percentile score = 62

Interquartile Range 6,62, 2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99 IQR = 75th percentile - 25th percentile 56 =

N = 40 Practice

N = = 2 40(.25) = 10

Practice Find the range for: 8, 4, 10, 15, 25, 56, 76, 64, 43, 4, 56, , 68.2, 78.3, 59.5, 78.6, 75.2, 12.9, , 51.25, 58.00, 96.34, 54.43

Practice Find the range for: 8, 4, 10, 15, 25, 56, 76, 64, 43, 4, 56, 22 –Range = = , 68.2, 78.3, 59.5, 78.6, 75.2, 12.9, 3.2 –Range = = , 51.25, 58.00, 96.34, –Range = = 51.33

Practice Find the interquartile range for: 8, 4, 10, 15, 25, 56, 76, 64, 43, 4, 56, , 68.2, 78.3, 59.5, 78.6, 75.2, 12.9, , 51.25, 58.00, 96.34, 54.43

Practice Find the interquartile range for: 8, 4, 10, 15, 25, 56, 76, 64, 43, 4, 56, 22 4, 4, 8, 10, 15, 22, 25, 43, 56, 56, 64, 76 (12).25 = = 48

Practice Find the interquartile range for: 8.5, 68.2, 78.3, 59.5, 78.6, 75.2, 12.9, , 8.5, 12.9, 59.5, 68.2, 75.2, 78.3, 78.6 (8).25 = = 69.8

Practice Find the interquartile range for: , 51.25, 58.00, 96.34, , 54.43, 58.00, 96.34, (5).25 = 1.25 ( )/2 = ( )/2 = = 46.62

Boxplots The boxplot graphically displays three different characteristics of the distribution –Extreme scores –Interquartile range –Median

Boxplot

Interquartile range 25th - 75th percentile

Boxplot Extreme Scores

Boxplot Median

Boxplot Skew -- Look at the “whiskers” to determine if the distribution is skewed

Create a boxplot Create a boxplot with this data set 2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99

Create a boxplot Create a boxplot with this data set 2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99 Median = 25th = 75th = Lowest = Highest =

Create a boxplot Create a boxplot with this data set 2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99 Median = th = 6 75th = 62 Lowest = 2 Highest = 99

Neuroticism

Extraversion

Conscientiousness

A B C D E Which distribution has a positive skew?

A B C D E Which distribution has a negative skew?

A B C D E Which distribution is most compact?

A B C D E Which distribution has a median close to 25?

A B C D E Which distribution is most symmetrical?

A B C D E Which distribution has has the largest range?

Review Measures of central tendency –Mean –Median –Mode Measures of variability –Range –IQR

Variability Range IQR Problem with range and IQR –variability is still measured with only two numbers!

Deviation Score Formula Deviation scores

= 16

Sample 1 vs. Sample 2 Sample 1: Raw scores:15, 12, 17, 20 Sample 2: Raw scores:26, 6, 1, 31

Sample 1 vs. Sample 2 Sample 1: Raw scores:15, 12, 17, 20 Deviation scores:-1, -4, 1, 4 Sample 2: Raw scores:26, 6, 1, 31 Deviation scores:10, -10, -15, 15

Deviation Scores As variability increases the absolute value of the deviation scores also goes up! How can we use this information to create a measure of variability?

How about?

Formula

- 1

Why “ – 1” ? Without it the answer is biased -- its answer tends to be too small Page 53 – 56 explain Don’t worry about why -- unless you want too!!

( )

= 6 ( )

= 6 ( )

= 6 ( )

= 6  = 38 ( )

Formula - 1

Formula

Formula

Formula

Formula

Practice For the sample data below calculate s 6, 8, 4, 3, 4, 5

() = 5  = 16

Practice