Chapter 4. Displaying and Summarizing. Quantitative Data

Transcription

1 STAT 141 Introduction to Statistics Chapter 4 Displaying and Summarizing Quantitative Data Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

2 4.1 Histograms 1 We divide the range of the data into classes of equal width. 2 Count the number of observations in each class. 3 Draw the histogram. Each bar has equal width and the height of each bar is the class count. Relative frequency histograms Replace the counts on the vertical axis with the percentage or proportion of the total number of cases falling in each class. (Step 2) Question? Compare histograms with bar charts. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

3 Example 4.1 Three steps to create a histogram for quantitative data. Figure: Quantitative data Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

4 Example 4.1 (cont d) 1: Find the range of the data: [ , ]. Consider a expended range: [1980,2024]. Use class width 4 to divide the range into 11 classes. Division is NOT unique. We normally choose integer width. 2: Count the number of observations in each class : Draw the histogram. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

5 Example 4.1 (cont d) Figure: Histogram of Example 4.1 Bin Zou STAT 141 University of Alberta Winter / 31

6 4.2 Stem-and-Leaf Displays 1: Divide each observed value into two parts: - The leading digit(s) of a number is called stem. - The rest of the digit(s) of a number is called leaf. Example: for a two-digit number 19, 1 is the stem and 9 is the leaf. Note: use only one digit for each leaf, either round or truncate the data values to one decimal place after the stem. 2: List the stems in a column (with the smallest at the bottom), and place a vertical line to the right of this column. 3: For each measurement, record the leaf portion in the same row as its corresponding stem. 4: Order the leaves from lowest to highest in each stem. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

7 Example 4.2 Construct a stem-and-leaf plot for the prices of walking shoes. Prices of walking shoes in $: First, we order the data from smallest to largest: The first digit of each observation is the stem, while the second digit is the leaf. The stems of the data are: 4, 6, 7, 8 and 9. Place them in a column. Put the leaves in each column with the ascending order. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

8 Stem-and-leaf Plot for Example Table: Stem-and-leaf Plot for Example 4.2 Remark: count all repeating leaves in each stem. order is important for both stems and leaves. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

9 More Questions about Stem-and-leaf Question 1 What if the data were 520, 541, 542,...? Or even in decimals, like 12.2, 14.8, 18.9,...? Use the first two digits as the stem, and the last one as the leaf, e.g., 52 0, Question 2 What if there are too many leaves for a stem? For instance, We can further divide the stem 5 into 5a (containing all observations from 50 to 54) and 5b (containing all observations from 55 to 59). Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

10 4.3 Dotplots A dotplot places a dot along an axis for each case in the data. Example 4.3 Assume the data are given by: Figure: Dotplot of Example 4.3 Note: Dotplots can be drawn vertically as well, meaning switch the axes in the figure above. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

11 4.4 The Shape of a Distribution Modes Modes are peaks (or humps) of a histogram. Equivalently, the mode is the value that occurs with the highest frequency in a data set. A histogram with one mode is called unimodal. Histograms with two peaks are bimodal. Those with three or more are called multimodal. A histogram with no mode (namely, all categories have approximately the same counts) is called uniform. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

12 Examples Figure: Histograms of different types Bin Zou STAT 141 University of Alberta Winter / 31

13 Symmetric Histograms When folded along a vertical line through the middle, a symmetric histogram should have the edges match pretty closely. Figure: Example of a symmetric histogram Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

14 Skewed Histograms If one tail stretches out farther than the other, the histogram is said to be skewed to the side of the longer tail. Figure: Example of skewed histograms In the above example, the histogram on the left (colored in blue) is skewed to the left (also called negatively skewed), and the one on the right is skewed to the right (positively skewed). Bin Zou STAT 141 University of Alberta Winter / 31

15 Outliers Outliers are extremely large or small observations, which are located away from the main body of the distribution. Apparently, the three observations in the leftmost bar are outliers. Bin Zou STAT 141 University of Alberta Winter / 31

16 Review of the Shape of A Distribution Example 4.4 Recall that there are three aspects regarding the shape of a distributions: number of modes, symmetry, and outliers. Discuss the shape of the distribution described in the picture below. Figure: The shape of a distribution Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

17 4.5 The Centre of a Distribution Measuring the Center: Mean Suppose a sample consists of n observations: x 1,x 2,...,x n. The mean (or average) of these values is given by x := n x i i=1 n = x 1 + x x n. n x is pronounced x-bar. The notation reads as sigma (Greek letter S ), meaning the summation over the index. Example 4.5 Given x 1 = 14.1, x 2 = 3.2, x 3 = 25.3, x 4 = 2.8,x 5 = 17.5,x 6 = 13.9, x 7 = Calculate the mean. x = x 1 + x x 7 7 = = Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

18 More about Mean Mean is a good measure of the centre when the data are (approximately) symmetric. If a distribution is skewed or has outliers, then mean is NOT a reliable measure of the centre. To see the impact of a single observation on the mean, we change x 7 in Example 4.5 from 45.8 to The new mean is calculated as follows x = = Compared with the mean calculated in Example 4.5, the new mean is more than 10 times bigger. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

19 Measuring the Centre: Median For a skewed distribution, the median is a better measure of the centre comparing with the mean. For a set of data in ascending (or descending) order, the median is the value that divide the data in half. Denote n the total number of observations. Assume that the data has been sorted in ascending order. If n is odd, then the median is the observation in the n+1 2 position. If n is even, then the median is the average of the two values in positions n 2 and n Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

20 Examples Example 4.6 The data are the same as in Example 4.5: x 1 = 14.1, x 2 = 3.2, x 3 = 25.3, x 4 = 2.8,x 5 = 17.5,x 6 = 13.9, x 7 = Rearrange the data in ascending order as follows: 17.5, 2.8, 3.2, 13.9, 14.1, 25.3, The total number of observations is 7, an odd number. Hence, the median is in the = 4th position. Namely, 13.9 is the median. Note: since the median is only related to the middle value(s), it is NOT affected by outliers. In the above example, if we change x 7 from 45.8 to 1000, the median stays the same. What happened to the mean example? Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

21 Examples Example 4.7 The previous example consists of odd number observations. Now, we add another observation with the value 35.7 to the previous data, and recalculate the median. The ordered data in Example 4.7 are listed as: 17.5, 2.8, 3.2, 13.9, 14.1, 25.3, 35.7, With n = 8, the median is the average of the values in the 4th and 5th positions. Hence, the median is given by = Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

22 Mean VS Median In a symmetric distribution, mean = median. In a positively (right) skewed distribution, mean > median. In a negatively (left) skewed distribution, mean < median. Example 4.8 Given the data: -2, -1, 0, 1, 2. The data set is symmetric about 0, and mean = median = 0. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

23 4.6 The Spread of a Distribution The more the data vary, the less a measure of centre can tell us. Why? Let s have a look up an extreme example. Assume you know the mean of the given data is 10, but the data are unknown to you. Both the sets (-10000, 30, 10000) and (10, 10, 10) have the same mean 10. However, these two sets of data are significantly different. We need to know the spread of a distribution as well. Range Range = max - min. Note: range defined here is a singe number, not an interval as in common sense. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

24 The Interquartile Range A percentile is a measure that indicates the value below which a given percentage of observations in a group of observations fall. The first quartile (also called lower quartile), Q1, is the percentile of 25%, meaning 25% percentage of observations are below Q1. The median (second quartile) is the percentile of 50%. Sometimes, we use Q2 to denote the median. Similarly, we define the third quartile (also called upper quartile), Q3, as the percentile of 75%. Interquartile range (IQR) = Q3 Q1. Note: IQR is a proper measure of spread when a distribution is skewed. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

25 Examples Example 4.6 revisited Data are ordered as: 17.5, 2.8, 3.2, 13.9, 14.1, 25.3, We have calculated Q2 = 13.9 in Example 4.6. Split the data into two equal parts: 17.5,2.8,3.2,13.9 and 13.9, 14.1, 25.3, Note: when n is odd (7 in this example), the median is included in both parts. Q1 is then the median of the first part of the data, calculated as Q1 = = 3.0. Q3 is the median of the second part of the data, calculated as Q3 = = Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

26 Examples Example 4.7 revisited Data are ordered as: 17.5, 2.8, 3.2, 13.9, 14.1, 25.3, 35.7, Separate the data into two parts with equal number. The set of the data has 8 observations, so each new part has 4 observations: 17.5, 2.8, 3.2, 13.9 and 14.1, 25.3, 35.7, Same as the example in the previous slide, we calculate the first quartile Q1 = = 3.0, 2 and the third quartile Q3 = = Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

27 The Standard Deviation If a distribution is skewed, use IQR. What if a distribution is (approximately) symmetric? The answer is: the standard deviation. Assume a data set of n observations: y 1,y 2,...,y n. The mean is denoted as ȳ. The formula to calculate the standard deviation is n (y i ȳ) 2 n y i=1 i 2 nȳ 2 i=1 s = =. n 1 n 1 s 2 is called the variance. Note: the standard deviation defined here is actually the standard deviation of a sample, NOT a population. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

28 Examples Figure: Finding the standard deviation Bin Zou STAT 141 University of Alberta Winter / 31

29 Empirical Rules There are three empirical rules about the standard deviation. (1) About 68% of the data will lie within 1 standard deviation of the mean. (2) Nearly all, about 95%, of the data will lie within 2 standard deviations of the mean. (3) Virtually all, about %, of the data will lie within 3 standard deviations of the mean. You will know the reason behind these rules after we have covered the Normal distribution. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

30 5-Number Summary The 5-number summary for a set of data reports: Minimum, Lower quartile (Q1), Median (Q2), Upper quartile (Q3), Maximum. The 5-number summary provides measures for centre, spread, and skewness. For a right skewed distribution, Q2 Q1 < Q3 Q2. For a left skewed distribution, Q2 Q1 > Q3 Q2. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31

31 Summarizations of Chapter 4 1 Displaying quantitative data: histogram, stem-and-leaf plot, dotplot. 2 Shape: unimodal/bimodal/multimodal/uniform, symmetric/skewed, outlier. 3 Symmetric distributions: mean + standard deviation. 4 Skewed distribution: median + IQR (5-number summary). Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31