MATH 2560 C F03 Elementary Statistics I Lecture 1: Displaying Distributions with Graphs. Outline.

Transcription

1 MATH 2560 C F03 Elementary Statistics I Lecture 1: Displaying Distributions with Graphs. Outline. data; variables: categorical & quantitative; distributions; bar graphs & pie charts:

2 What Is Statistics? Statistics is the science of collecting, organizing, and intrepreting numerical facts, which we call data. Individuals: A data set contains information on a collection of individuals: people, animals, things, etc. The data for one individual make up a case. What are Data? Data Variables: For each individual, the data give values for one or more variables. A variable describes some characteristic of an individual, such as a person s height, gender, or salary. What are Variables? Categorical: Some variables are categorical (place each individual into a category, such as male or female) Variables Quantitative: others variables are quantitative (have numerical values that measure some characteristic of each individual, such as height in cm or annual salary in dollars). CyberStat Corporation Data about its Employees 0 A: B: C: D: E: F: 1 Name Age Gender Race Salary Job Type 2 Fleetwood, Delores 39 Female White 62,100 Management 3 Perez, Juan 27 Male White 47,350 Technical 4 Wang, Lin 22 Female Asian 18,250 Clerical 5 Johnson, LaVeme 48 Male Black 77,600 Management Each row records data on one individual. Each row of data calles a case. Each column contains the values of one variable for all the individuals. There are 5 variables: gender+ race+job type=3 are categorical variables, and age+salary=2 in dollars are quantitative variables. Exploratory data analysis uses graphs and numerical summaries to describe the variables in a data set and the relations among them.

3 Distributions of Categorical Variables. The distribution of a variable tells us what values it takes and how often it takes these values. Bar graphs display the distributions of categorical variables with bars Categorical Variables Pie charts display the distributions with pie charts. Distribution of Marital Status for All Americans Age 18 and Over Marital Status: Count (millions): Percent: Never married Married Widowed Divorced

4 Stemplots and Histograms. Stemplots and histograms display the distributions of quantitative variables. Stemplots. Stemplots separate each observation into a stem and one-digit leaf. How to make a stemplot? To make a stemplot: 1. Separate each observation into a stem consisting of all but the final (rightmost) digit and a leaf, the final digit. Stems may have as many digits as needed, but each leaf contains only a single digit. 2. Write the stems in a vertical column with the smallest at the top, and draw a vertical line at the right of this column. 3. Write each leaf in the row to the right of its stem, in increasing order out from the stem. Examples. Example 1.4.a. Babe Ruth s Home Runs hits of 15 Years

5 Stemplot of the Data in Example 1.4.a (a) (b) (c) (a) write the stems; (b) go through the data writing each leaf on the proper stem; (c) arrange the leaves on each stem in order out from the stem. Example 1.4.b Roger Maris home runs hits of 10 years Back-to-Back Stemplot. Stemplot Comparing of the Data of the Exs 1.4.a and 1.4.b Ruth Maris

6 Example 1.5 A marketing consultant observed 50 consecutive shoppers at a supermarket. One variable of interest was how much each shopper spent in the store. Below, there are the data (in dollars), arrangedin increasing order: Supermarket Spending Figure 1.3 (a). Stemplot without splitting the stems (Example 1.5.)

7 Stemplot with splitting the stems (Example 1.5.)

8 Histograms. A histogram breaks the range of values of a variable into intervals and displays only the count or percent of the observations that fall into each interval. Histograms plot the frequencies or relative frequencies of classes of values. Example 1.9. Examining the performance of 947 Gary, Indiana, seventh graders on the reading portion. Their vocabulary scores, expressed as an equivalent grade level, ranged from 2.0 to The score of the first few of the 947 student are: 5.4, 6.8, 2.0, 7.6, 6.6, 7.8, 8.1, 5.6, 6.0, 7.9, 2.9,... How to make a Histogram? To make a histogram proceed as follows: 1. Divide the range of the data into classes of equal width. 2. Count the number of observations in each classes. These counts are called frequencies, and a table of frequencies for all classes is a frequency table. 3. Draw the histogram. To make a histogram of the distribution of scores, proceed as follows: 1. Divide the range of the data into classes of equal width. In this case,it is natural to use grade levels, so the classes are: [2.0, 3], [3.0, 4.0],..., [12.0, 13.0]. 2. Count the number of observations in each classes. Table 1.2 is a frequency table of the Gray vocabulary scores. 3. Draw the histogram. In the histogram in Figure 1.4, the4 vocabulary score scale is horizontal and the frequency scale is vertical.

9 Table 1.2 for the Data of Example 1.9. Vocabulary Scores for Seventh Graders, Gary, Indiana Class: Number of Students: Percent: Total Frequencies (counts) or relative frequencies (percents or fractions) are summarized a large number of observations on a single variable in a table.

10 Examining a Distribution. Look for shape, center, and spread and for clear deviations from the overall shape to examine a distribution. The center of a distribution is described by its midpoint, the value with roughly half the observations taking smaller values and half taking larger values. The spread of a distribution is described by giving the range between the smallest and largest values. The shape is described by one or several major peaks, called modes, and by symmetric or skewed in one direction form. A distribution with one major peak is called unimodal. A distribution is symmetric if the values smaller and larger than its midpoint are mirror images of each other. A distribution is skewed to the right if the right tail (larger values) is much longer than the left tail (smaller values). Outliers are observations that lie outside the overall pattern of a distribution. Example 1. The distribution of Babe Ruth s home run counts is symmetric and unimodal. The midpoint is 46 home runs, and the range (spread) is from 22 (smallest value) to 60 (largest value). There are no outlier. In particular, Ruth s record 60 home runs in one season does not stand outside his overall pattern. Example 2. Maris s 61 home run season, on the other hand, is an outlier that stands far above his other years. Maris s peformance (without the outlier) is roughly symmetric with midpoint 23 and range 8 to 39. Example 3. The distribution of supermarket spending (Figure 1.3(a)) is skewed to the right. The direction of the long tail (to the right) determines the direction of skewness. Because the largest amounts do not fall outside the overall skewed pattern, we do not call them outliers. The midpoint of the distribution (count up 25 entries in the stemplot) is 28D. The range is 3 to 93D. The distribution is unimodal. The right-skewed shape in Figure 1.3. is common among distributions of money amounts.

11 Table 1.1. Newcomb s measurements of the passage time of light

12 Time Plots A time plot is the plot for observations on a variable which are taken over time. To make a plot graph time horizontally and the values of the variable vertically. A time plot can reveal trends or other changes over time such as seasonal variation.

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