STAT 211 – 019 Dan Piett West Virginia University Lecture 1

Overview 1.1 Science of Statistics 1.2 Displaying Small Sets of Numbers 1.3 Graphing Categorical Data BREAK - Problems 1.4 Frequency Histograms 1.5 Density Histograms 1.6 Centers of Data 1.7 Mean vs Median

Section 1.1 Science of Statistics

Definitions Statistics The science of collecting, organizing, and analyzing data for the purpose of estimation and making inferences. Data Values which arise from observing characteristics on a selected group (sample) of individuals. The characteristic(s) which are observed are called variables. Population The entire group of “things” from which data may be collected. Sample The selected group of the population that is studied.

Population vs. Sample Population All undergraduate students at WVU Sample A group of 100 undergraduate students at WVU All WVU Students (population) Group of 100 Students (sample)

Types of Data Variables The characteristics which we look at when observing data. These can be classified into four groups. Numeric Discrete Continuous Categorical Ranked (Ordinal) Unranked (Nominal)

Numeric Variables Variables whose values represent quantities. This can be further broken down into discrete and continuous. Discrete Numeric Variables which usually arise by counting Examples Number of cars in a parking lot Number of courses taken in a semester Continuous Numeric Variables which arise by measuring Examples Time spent running a marathon Weight

Categorical Variables Variables that are not numeric. These can be further classified into ranked and unranked. Ranked Categorical Variables If the possible values of a categorical value follow a “natural” ordering Example Olympic Medal Unranked Categorical Variables Anything that is not ranked Example Blood Type

Two Major Branches of Statistics Descriptive StatisticsInferential Statistics Use graphical displays and numeric summarizations to represent data. First half of this semester Use analytic methods and theory of probability to draw conclusions or make decisions. Second half of this semester

Section 1.2 Displaying Small Sets of Numbers

Displaying Small Sets of Data We will be looking at three different ways to display small sets of data. Dot Plots Stem and Leaf Display Histograms

Dot Plot Each Dot represents one data value Example – Height of Students 62, 71, 65, 68, 64, 72, 66, 68, 70, 67, 67, 68, 64, 65, 68

Stem and Leaf Display Represents the data using the actual digits that make up the data. Leading digit becomes the stem Trailing digit becomes the leaf

Stem and Leaf Examples Example – Exam Grades 76, 74, 82, 96, 66, 76, 93, 86, 84, 62, 82, 75, 58, 71, 73, 79, 65, 80 Example - Decimals 1.3, 2.4, 1.7, 3.2, 5.6

Outliers Outlier An observation whose value is unusual or extreme. Example 15, 21, 13, 18, 23, 19, 26, 16, 71, 22, 14, is an outlier

Section 1.3 Graphing Categorical Data

Grouped Frequency Table Frequencies are tabulated for each value of the categorical variable. Example – STAT 101 Grade Distribution (2000)

Bar Graph Represents categorical data by showing the amount of data that belong to each category as proportionally sized rectangles. Example: US Coal Production (in millions) WV – PA – KY – WY – Other – 120.4

Pie Chart (Circle Graph) A pie chart shows the amount of data that belongs to each category as a proportional part of a circle. Example: US Coal Production (in millions) WV – PA – KY – WY – Other – 120.4

Section 1.4 Frequency Histograms

Grouped Frequency Distributions A list (or table) which pairs ranges for values of a variable with their frequencies (counts). Each range is called a class. Rules for constructing a grouped frequency distribution 1. Each class should be the same width. 2. Classes should not overlap. 3. Each observation falls into one and only one class. 4. Use between 3 and 15 classes.

Frequency Histogram A graphical representation of a frequency distribution Example – Math3 Exam Scores

Shape of a Distribution Symmetric Shaped Mound Shaped U-Shaped Uniform

Shape of a Distribution Asymmetric Shaped Skewed Right (Positive) Skewed Left (Negative)

Break

Section 1.5 Density Histograms

Density Histogram Relative Frequency Histogram A histogram in which the vertical axis represents percentages or proportions, rather than counts. Example – Math3 Exam Scores

Constructing Density Histograms 1. For each class, compute the percent of observations in each class. 2. Divide the percent associated with each class by the width of that class. This yields the percent of observations associated with each unit of the measurement scale. 3. Draw the histogram using the values computed in step 2.

More on Density Histograms If the bars of a histogram are arranged from largest to smallest, it is called a Pareto Chart. A density histogram is a histogram whose vertical axis is scaled so that the sum of the areas of it’s rectangles is 1 square unit. If the sample size, n, is very large, a density histogram can be used to estimate the distribution of the population from which the data was obtained.

Section 1.6 Centers of Data

Averages Average An average is a single value which represents all of the data. Types of Averages Sample Mean Sample Median Sample Mode

Sample Mean (x) Sample Mean Formula Example x: 14, 23, 8, 19, 41 Sample Mean = ( )/5 = 105/5 = 21

More on Sample Means If we acquire data from all members of a population, we can compute the population mean ( ) Because of this, we can use the sample mean to estimate the population mean

Sample Median ( ) Sample Median The middle value of the observed data values ranked from lowest to highest If the sample contains n observations, the median is the ½(n+1) ranked value Example: x: 14, 23, 8, 19, 41 Ranked: 8, 14, 19, 23, 41 n = 5 Position of the median: ½(5+1) observation = 3 rd Median = 19

Sample Median Even Example In the previous example, n was odd Example with an even n: x: 43, 26, 37, 19, 52, 80 Ranked: 19, 26, 37, 43, 52, 80 n=6 Position of the Median: ½ (6+1) observation =3½ position Median is the mean of the 3 rd and 4 th ranked observation Median = (37+43)/2 = 40

Sample Mode The observed value which occurs with the greatest frequency Example: x: 4, 3, 1, 4, 0, 3, 3, 1, 4, 0, 1, 1, 2, 2 Mode = 1

More Mode Examples A set of x might have no mode (every value occurs once) Mode = None NOT Mode = 0 A set of x might have more than one mode Mode = 5 and 6

Section 1.7 Mean vs. Median

Best Measure of the Center MeanMedian Use for approximately symmetric distributions Greatly influenced by outliers Use for significantly skewed distributions Not greatly influenced by outliers

How to Handle Outliers If the outlier is a “mistaken” data value, eliminate it from the analysis If the outlier is an actual data value, do not eliminate it

End of Class