Descriptive Statistics Healey Chapters 3 and 4 (1e) or Ch. 3 (2/3e) Measures of Central Tendency And Dispersion

Measures of Central Tendency 1. Mode = can be used for any kind of data but only measure of central tendency for nominal or qualitative data. Formula: value that occurs most often or the category or interval with highest frequency. Note: Omit Formula 3.1 Variation Ratio in Healey and Prus 2nd Cdn.

Example for Nominal Variables: Religion frequency cf proportion % Cum% Catholic 17 17 .41 41 41 Protestant 4 21 .10 10 51 Jewish 2 23 .05 5 56 Muslim 1 24 .02 2 58 Other 9 33 .22 9 80 None 8 41 .20 20 100 Total 41 1.00 100% Central Tendency: MODE = largest category = Catholic

Central Tendency (cont.) 2. Median = exact centre or middle of ordered data. The 50th percentile. Formula: Array data. When sample even #, median falls halfway between two middle numbers. To calculate: find(n/2)and (n/2)+1, and divide the total by 2 to find the exact median. When sample is odd #, median is exact middle (n+1) /2)

Example for Raw Data: Suppose you have the following set of test scores: 66, 89, 41, 98, 76, 77, 69, 60, 60, 66, 69, 66, 98, 52, 74, 66, 89, 95, 66, 69 1. Array data: 98 98 95 89 89 77 76 74 69 69 69 66 66 66 66 66 60 60 52 41 N = 20 (N is even)

To calculate: - find middle numbers(n/2)+(n/2 )+1 - add together the two middle numbers - divide the total by 2 First middle number: (20/2) = the 10th number 2nd middle number: (20/2)+1 = the 11th Look at data: the middle numbers are 69 and 69 The median would be (69+69)/2 = 69

Median for Aggregate (grouped) Data This formula is shown in Healey 1st Cdn Edition but NOT in 2/3 Cdn We will NOT COVER this one!

Properties of median: - for numerical data at interval or ordinal level -"balance point“ -not affected by outliers -median is appropriate when distribution is highly skewed.

3. Mean for Raw Data The mean is the sum of measurements / number of subjects Formula: (X-bar) = ΣXi / N Data (from above): 66, 89, 41, 98, 76, 77, 69, 60, 60, 66, 69, 66, 98, 52, 74, 66, 89, 95, 66, 69

Example for Mean Formula: = ΣXi / N = 1446 / 20 = 72.3 The mean for these test scores is 72.3

Mean for Aggregate (Grouped) Data (Note: not in text but covered in class) To calculate the mean for grouped data, you need a frequency table that includes a column for the midpoints, for the product of the frequencies times the midpoints (fm). Formula: = Σ (fm) N

Frequency table: Score f m* (fm) 41-50 1 45.5 45.5 51-60 3 55.5 166.5 41-50 1 45.5 45.5 51-60 3 55.5 166.5 61-70 8 65.5 524 71-80 3 75.5 226.5 81-90 2 85.5 171 91-100 3 95.5 286.5 N = 20 Σ (fm) = 1420 * Find midpoints first

Calculating Mean for Grouped Data: Formula: = Σ (fm) N = 1420 / 20 = 71 The mean for the grouped data is 71.

Properties of the Mean: - only for numerical data at interval level - "balance point“ - can be affected by outliers = skewed distribution - tail becomes elongated and the mean is pulled in direction of outlier. Example… no outlier: $30000, 30000, 35000, 25000, 30000 then mean = $30000 but if outlier is present, then: $130000, 30000, 35000, 25000, 30000 then mean = $50000 (the mean is pulled up or down in the direction of the outlier)

NOTE: When distribution is symmetric, mean = median = mode For skewed, mean will lie in direction of skew. i.e. skewed to right, mean > median (positive skew) skewed to left, median > mean (negative skew)

Measures of Dispersion Describe how variable the data are. i.e. how spread out around the mean Also called measures of variation or variability

Variability for Non-numerical Data (Nominal or Ordinal Level Data) Measures of variability for non-numerical nominal or ordinal) data are rarely used We will not be covering these in class Omit Formula 4.1 IQV in Healey and Prus 1st Canadian Edition Omit Formula 3.1 Variation Ratio in Healey and Prus 2/3 Canadian Edition

2. Range (for numerical data) Range = difference between largest and smallest observations i.e. if data are $130000, 35000, 30000, 30000, 30000, 30000, 25000, 25000 then range = 130000 - 25000 = $105000

Interquartile Range (Q): This is the difference between the 75th and the 25th percentiles (the middle 50%) Gives better idea than range of what the middle of the distribution looks like. Formula: Q = Q3 - Q1 (where Q3 = N x .75, and Q1 = N x .25) Using above data: Q = Q3 - Q1 = (6th – 2nd case) = $30000-25000 =$5000 The interquartile range (Q) is $5000.

3. Variance and Standard Deviation: For raw data at the interval/ratio level. Most common measure of variation. The numerator in the formula is known as the sum of squares, and the denominator is either the population size N or the sample size n-1 The variance is denoted by S2 and the standard deviation, which is the square root of the variance, by S

Definitional Formula for Variance and Standard Deviation: Variance: s2 = Σ (xi - )2 / N S.D.: s = A working formula (the one you use) for s.d is: 1 N ∑ Xi2 - ( ∑ Xi ) 2 N

Example for S and S2 : Data: 66, 89, 41, 98, 76, 77, 69, 60, 60, 66, 69, 66, 98, 52, 74, 66, 89, 95, 66, 69 Find ∑ Xi2 : Square each Xi and find total. Find (∑ Xi)2 : Find total of all Xi and square. Substitute above and N into formula for S. For S2 , simply square S. S = 14.76 S2 = 217.91

Another working formula for the standard deviation: Note that the definitional formula for s.d. is not practical for use with data when N>10. The working formulae should be used instead. All three formulae give exactly the same result.

Properties of S: always greater than or equal to 0 the greater the variation about mean, the greater S is n-1 (corrects for bias when using sample data.) S tends to underestimate the population s.d. so to correct for this, we use n-1. The larger the sample size, the smaller difference this correction makes. When calculating the s.d. for the whole population, use N in the denominator.

NOTE: σ, N and Mu (µ) denote population parameters s, n, x-bar ( ) denote sample statistics

Remember the Rounding Rules! Always use as many decimal places as your calculator can handle. Round your final answer to 2 decimal places, rounding to nearest number. Engineers Rule: When last digit is exactly 5 (followed by 0’s), round the digit before the last digit to nearest EVEN number.

Homework Questions Healey and Prus 1e: #3.1, #3.5, #3.11 (compute s for 8 nations also), #3.15 SPSS: Read the SPSS sections for Ch. 3 and 4 in 1st Cdn. Edition and for Ch. 4 in 2/3 Cdn. Edition Try some of the SPSS exercises for practice