1.1 Descriptive Statistics

1.1 Descriptive Statistics Numerical Data Data Categorical Data Example Suppose the following are the exam scores for the students in a class: 90 85 80 76 82 95 88 71 72 95 Each item is called an observation. Since each observation is a number, the data is numerical data. Example A survey asks voters about their political affiliation (Democrat, Republican, and Independent). The following is the result of 10 voters: D R R I R D D R R D Each observation is a category, so the data is categorical data. Example A survey asks people whether they eat at McDonald’s or not. 60 people says “Yes” and 29 people says “No”. This is categorical data because it is a summary of the following: Yes, ……, Yes (60 counts), No, ……, No (29 counts) Numerical data: Sample mean, sample variance and sample standard deviation For a numerical data set, the mean (or average) is defined as: x = ∑ of observations Number of observations = ∑ x i n Example The following are the quiz scores of a student: 9 7 8.5 7.5 8 Find the sample mean (or average) of the scores. Solution x = ∑ x i n = 9 + 7 + 8.5 + 7.5 + 8 5 = 40 5 = 8

The following are the formulas for sample mean, sample variance and sample standard deviation: Samplemean x = ∑ x i n Degree of freedom df = n − 1 Samplevariance s 2 = ∑ ( x i − x ) 2 df Samplestandard deviation s = √ s 2 Example The following are the quiz scores of a student: 9 7 8.5 7.5 8 Find the sample mean, sample variance and sample standard deviation. Solution x = ∑ x i n = 9 + 7 + 8.5 + 7.5 + 8 5 = 40 5 = 8 df = n − 1 = 5 − 1 = 4 s 2 = ∑ ( x i − x ) 2 df = ( 9 − 8 ) 2 + ( 7 − 8 ) 2 + ( 8.5 − 8 ) 2 + ( 7.5 − 8 ) 2 + ( 8 − 8 ) 2 4 = 1 + 1 + 0.25 + 0.25 + 0 4 = 2.5 4 = 0.625 s = √ s 2 = √ 0.625 = 0.79 Categorical data: Proportion (or percent) For categorical data, the sample proportion or percentage is defined as: Proportion = Part Total Example A survey asks voters about their political affiliation (Democrat, Republican, and Independent). A sample of 10 voters is selected with the following result: D R R I R D D R R D Find the proportion of voters in the sample who are Republican. Solution