Chapter 1 Data Presentation Statistics and Data Measurement Levels Summarizing Data Symmetry and Skewness

Statistics and Data Statistics – collection of techniques used in analyzing data – numbers produced in the analysis (eg. Average) Data – collection of measurements made on a number of subjects. Subjects – where information are drawn -experimental units

Data are usually stored in a row-and-column display called a spreadsheet. From page 2 of the textbook Row represents a subject and columns represent measure of variables.

Measurement Levels of Data Categorical Data – variables that yield categorical data Nominal – possible values are just names of categories – no apparent ordering between the possible values examples: Gender, Major, College Ordinal – there is an obvious ordering of the possible values example: Year level (Freshman, Sophomore …), Military ranking Numerical Data - variables that yield numerical data Interval – Interval exists but not ratios – zero does not mean absence of that variable examples: Temperature, IQ 60 F vs 30 F, there is 30 degrees difference between the two temperatures but it does not mean that 60 F is twice as warm as 30F Ratio – ratio exist examples: Age, Height, Number of classes taken this semester Types of Data

Ratio : there are 2 other levels under ratio Discrete: result of a counting process example: number of classes being taken, number of students in a class Continuous: result of a measuring process. example: height, age, weight, velocity

Summary of Data type and Levels

Summarizing Data Summarizing Categorical Data 1.Relative Frequency Table - represents the frequency of each type of categorical variable 2.Bar Chart - plot of the relative frequency table; order of categories is arbitrary 3.Pie Chart - also a plot of the relative frequency table, except in a circular shape

Relative Frequency Table

Bar Chart of the Relative Frequency Table Using Frequency Using Relative Frequency

Pie Chart

Summarizing Numerical Data Stem and Leaf Plot Relative Frequency Table and Histogram similar concept with the categorical data determine the following: number of classes, class width For example: MIN, MAX, number of classes, width = (MAX -MIN) /(classes-1) The intervals in each class should be mutually exclusive. The histogram will just be the graphical presentation of the RTF Box-and-Whisker Plot a graphical picture of the distribution of quarters of the data. Useful for comparing distributions of two or more variables Minimum Q1 (first quartile) – the upper boundary of the first quarter Median – divides the data into lower and upper halves. Q3 (third quartile) – the upper boundary of the third quarter Maximum Dotplot similar to the histogram but used for moderately large data this can also be used in studying outliers in the data

Stem-and-leaf Display Summer 2 Quiz Data: 8, 11, 13, 19, 21, 23, 25, 25, 25, 28, 31, 35, 39, 47 Stemplot of Summer 2 Quiz

Relative Frequency Table and Histogram Summer 2 Quiz Data: 8, 11, 13, 19, 21, 23, 25, 25, 25, 28, 31, 35, 39, 47 For example, 4 classes is desired. MIN=8, MAX=47 Class width = (47-8)/(4-1)=39/3=13 Note: intervals include the right endpoint but not the left endpoint.

Histogram of the Summer 2 Quiz Data

Boxplot or Box-and-Whisker Plot Minimum = 8 Q1 (first quartile) =19 Median = 25 Q3 (third quartile) = 31 Maximum = 47 Summer 2 Quiz Data: 8, 11, 13, 19, 21, 23, 25, 25, 25, 28, 31, 35, 39, 47

Symmetry and Skewness Examining symmetry and skewness determines the shape of the data If the left tail is longer than the right tail, then the data is left-skewed. If the right tail is longer than the left tail, then the data is right-skewed. If the left tail is almost the same as the right tail, then the data is symmetric. Stem-and-leaf display, Histogram and Boxplot can be used to examine symmetry and skewness.

The left tail is longer than the right tail, hence the data is left-skewed.