# The Big Picture. Describing Data: Categorical and Quantitative Variables Population. Descriptive Statistics. Community Coalitions (n = 175)

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1 Describing Data: Categorical and Quantitative Variables Population The Big Picture Sampling Statistical Inference Sample Exploratory Data Analysis Descriptive Statistics In order to make sense of data, we need ways to summarize and visualize it. Community Coalitions (n = 175) Summarizing and visualizing variables and relationships between two variables is often known as exploratory data analysis (also known as descriptive statistics). The type of summary statistics and visualization methods to use depends on the type of variables being analyzed (i.e., categorical or quantitative). One Categorical Variable What is your race/ethnicity? White Black Asian Other Display the number or proportion of cases that fall into each category. Frequency Table A frequency table shows the number of cases that fall into each category: What is your race/ethnicity? White Black Asian Other Total

2 Proportion Proportion The sample proportion ( ) of directors in each category is number of cases in category pˆ total number of cases White Black Asian Other Total The sample proportion of directors who are white is: % 175 Proportion and percent can be used interchangeably. Relative Frequency Table A relative frequency table shows the proportion of cases that fall in each category. Bar Chart In a bar chart, the height of the bar corresponds to the number of cases that fall into each category. White Black Asian Other All the numbers in a relative frequency table sum to White Black Asian Other Pie Chart Two Categorical Variables In a pie chart, the relative area of each slice of the pie corresponds to the proportion/percentage in each category. Look at the relationship between two categorical variables 1. Race/Ethnicity 2. Gender Black 17% White 63% 17% Asian 1% Other 2% 2

3 It doesn t matter which variable is displayed in the rows and which in the columns. What proportion of female directors are? A. 12/29 B. 12/175 C. 12/81 D. 81/175 E. 29/175 What proportion of directors are female? A. 12/29 B. 12/175 C. 12/81 D. 81/175 E. 29/175 The proportion of female directors that are The proportion of directors that are female What proportion of directors are female and? A. 12/29 B. 12/175 C. 12/81 D. 81/175 E. 29/175 Side-by-Side Bar Chart In a side-by-side bar chart, the height of each bar corresponds to the number of cases that fall into each category of the table Female Male White Black Other 3 1 3

4 Side-by-Side Bar Chart In a side-by-side bar chart, the height of each bar corresponds to the number of cases that fall into each category of the table Segmented Bar Chart A segmented bar chart is like a side-by-side bar chart, but the bars are stacked instead of side-by-side White Black Other 1 8 Other Black White Female Male Female Male Difference in Proportions A difference in proportions is the difference in proportions for one categorical variable (e.g., the proportion who are ) calculated for the different levels of another categorical variable (e.g., gender) Difference in Proportions What is the difference in proportion of male directors who are and female directors who are? M H = sample proportion of male directors who are F H = sample proportion of female directors who are Difference in Proportions = MH - FH What is the difference in gender proportions among directors? M H - F H The proportion of male directors who are The proportion of female directors who are 17/94 12/81.33 One Quantitative Variable When describing quantitative variables we are interested in the distribution of the values it s shape, center, and spread. Shape: Form of the distribution of values Center: Main peak Spread: Relative deviation of the values To understand these concepts we ll look at quantitative variables from the student survey. 4

5 Dotplot Histogram In a dotplot, each case is represented by a dot and dots are stacked. Number of text messages sent yesterday Create bins (i.e., value intervals) and place each case in the appropriate bin based on its value for the variable of interest. The height of the each bar corresponds to the number of cases that have values falling within that particular interval. Bar Charts vs. Histograms Shape Although they look similar, a histogram is not the same as a bar chart. A bar chart is for categorical data, and the x-axis has no numeric scale. A histogram is for quantitative data, and the x-axis is numeric. Long right tail For a categorical variable, the number of bars equals the number of categories, and the number in each category is fixed. For a quantitative variable, the number of bars (or bins) in a histogram is up to you, and the appearance can differ with different number of bars. Symmetric Right-Skewed Left-Skewed Measures of Center Notation The sample size, the number of cases in the sample, is denoted by n. A variable is often denoted by x, and x 1, x 2,, x n represent the n values of the variable x. Example: x = The number of body piercings x 1 = 6 x 2 = x 3 = 1 x 4 = 5 5

6 Measures of Center Mean The sample mean ( ) is the average, and is computed by adding up all the numbers and dividing by the number of cases. Sample Mean: x x x x n i n n i 1 1 n Measures of Center Median The sample median (m) is the middle value when the data is ordered. If there are an even number of values, the median is the average of the two middle values. Outliers An outlier is a value that is notably different from the other values (e.g., much larger or smaller than the other values) Average number of times Number of text messages checking Facebook per day sent yesterday Resistance A statistic is resistant if it is not heavily affected by outliers. The median is resistant, the mean is not resistant. Number of text messages sent per day: Mean Median Outlier Included Outlier Removed Outliers When calculating statistics that are not resistant to outliers, look for outliers and decide whether the outlier is a mistake. If not, you have to decide whether the outlier is part of your population of interest or not. Measures of Center Distribution of NHL Player Salaries m = \$1,25, = \$2,21, Mean is pulled in the direction of skewness Usually, for outliers that are not a mistake, it s best to run the analysis twice, once with the outlier(s) and once without, to see how much the outlier(s) affect the results Salary (in millions) 6

7 Assignment Getting Variable Statistics from the GSS Part I Graded Problems 2.18 and 2. Additional Practice Problems (not to be turned in): 2.11 and 2.57 Part II Goto Find 3 categorical variables and provide the proportion for each category for each variable. Find 3 quantitative variables and provide the mean & median and whether the distribution of values are symmetric, right skewed, or left skewed. Enter the variable name here. Make sure these boxes are checked. If applicable select the appropriate type of chart. Click on this button and the variable statistics will open up in a new window. Summary Statistics Proportion Frequency table Relative frequency table Visualizations Bar chart Pie chart Summary: One Categorical Variable Summary: Two Categorical Variables Summary Statistics Two-way table Difference in proportions Visualizations Side-by-side bar chart Segmented bar chart 7

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