Chapter 1 Data PresentationStatistics and Data

Measurement LevelsSummarizing 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 dataNominal – 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 dataInterval – Interval exists but not ratios

– zero does not mean absence of that variableexamples: Temperature, IQ60 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 processexample: 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

Major Frequency Rel. Freq. ACT 16 0.29GBS 28 0.5MGT 8 0.14MKT 4 0.07Total 56 1

Bar Chart of the Relative Frequency Table

0

5

10

15

20

25

30

ACT GBS MGT MKT

Frequency

Frequency

Using Frequency

Using Relative Frequency

0.29

0.5

0.14

0.07

0

0.1

0.2

0.3

0.4

0.5

0.6

ACT GBS MGT MKT

Percent Distribution of Major

Rel. Freq.

Pie Chart

0.29

0.5

0.14

0.07

Percent Distribution of Major

ACT

GBS

MGT

MKT

Summarizing Numerical Data

Stem and Leaf PlotRelative Frequency Table and Histogram

similar concept with the categorical datadetermine the following: number of classes, class widthFor 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

MinimumQ1 (first quartile) – the upper boundary of the first quarterMedian – divides the data into lower and upper halves.Q3 (third quartile) – the upper boundary of the third quarterMaximum

Dotplot

similar to the histogram but used for moderately large datathis 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

0 81 1 3 92 1 3 5 5 5 83 1 5 94 7

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=47Class width = (47-8)/(4-1)=39/3=13

Class Freq. Rel.Freq-5 - 8 1 7%8 - 21 4 29%21 - 34 6 43%34 - 47 3 21%

14 100%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 = 8Q1 (first quartile) =19Median = 25Q3 (third quartile) = 31Maximum = 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.