ExploringData

Graphing and Summarizing Univariate Data

Graphing the Data• Graphical displays of quantitative data

include:▫ Dotplot▫ Stemplot▫ Histogram▫ Cumulative Frequency Plots (ogives)▫ Boxplots

Dotplot

• As you might guess, a dotplot is made up of dots plotted on a graph.

• Each dot can represent a single observation from a set of data, or a specified number of observations from a set of data.

• The dots are stacked in a column over a category or value, so that the height of the column represents the frequency of observations in the category.

Dotplot ExampleNumber of Dogs in Each Home in My Block

** ** * ** * * *0 1 2 3# of Dogs

Stemplot Stems Leaves 15 1 14 13 12 2 6 11 4 5 7 9 10 1 2 2 2 5 7 9 9Key: 9 0 2 3 4 4 5 7 8 9 9

15 1 = 151 8 1 1 4 7 8

Key: 110

7 represents an IQ score of 117

Histogram

Note bars touch andvariable is quantitative

Cumulative Frequency PlotTypical Wait Times

Wait Times ( in Hrs.)

CumFreq (%)

Often Used for estimating medians, quartiles, & Percentiles

Boxplot

Med MaxMin 1Q 3Q

Based on 5- Number Summary

SHAPES of Boxplots• Previous was symmetric• Below is Skewed left

• Below is Skewed Right

Checking for outliers

An outlier is any value that is either

• greater than Q3 + 1.5*IQROR

• less than Q1 – 1.5*IQR

Note that whiskers always end at a data value

What Is Required on ALL Plots?

• Title• Labels on the horizontal and vertical axes

- be sure if you are using 3 to represent 3,000 that that information is in the label

• Scales on both axes (sometimes this is not needed, for example on boxplots)

• Labels for each plot if the graph includes multiple data sets (e.g. parallel boxplots)

How to Describe the GraphsUse your SOCS:o S hapeo O utliers and/or other unusual featureso C entero S pread

Discuss all characteristics IN CONTEXT.

Shape• Four Basic Shapes:• Symmetric

• Uniform

• Skewed left or skewed toward small values

• Skewed right or skewed toward large values

Should I Say Normal? Be careful when you describe the shape of

a mound-shaped, approximately symmetric distribution. The distribution may or may not be normal. Graders will accept the description as approximately normal, but they will not accept that the distribution is normal based only on a mound-shaped, symmetric graph.

Outliers and other Unusual Features

The Usual Unusuals:

• Gaps

• Clusters

• Outliers

• Peaks – ex. Bimodal

Center• Mean and median are both measures of

center • Median – put the values in order and the

median is the middle value (or the mean of the two middle values) – the median divides a histogram into two equal areas

• Mean – add the values and divide by the number of values you have – the mean is the balance point for a histogram

SpreadSeveral ways to describe:• Range – calculate max - min; the range

gives you the total spread in the data.• IQR – calculate Q3 – Q1; IQR gives you

the spread of the middle 50% of the data• Standard deviation – the average distance

of data values from the mean

How Does the shape impact Mean and Median?

• If the shape is approximately symmetric, the mean and median are approximately equal.

• If the shape is skewed, the mean is closer to the tail than the median.

Ex. Salaries – the mean will be larger than the median because salaries are usually skewed right

The Converse May Not Be True

Be careful –

If the mean is not equal to the median, you cannot conclude automatically that the shape is skewed.

Comparing Graphs Means to Compare – not just list characteristics

• Okay to sayo The mean of x= 8 is less than the mean

of y = 9.o The medians of x and y are about the same.o The median of x is slightly larger.o The shapes are both skewed left.

• Not Okayo The mean of x is 8 and the mean of y is 9.oMedian x = 4, median y =4.o The shapes are similar.

When Do You Use X-Bar/Sx and When Do You Use the 5-Number

Summary?• If the distribution is symmetric, use mean and

standard deviation.• If the distribution is skewed, use the 5-number

summary.

• Note that the mean and standard deviation are not resistant to outliers; the median and IQR are resistant.

Other Key Locations on Distributions• Percentile – the smallest value x for which n

percent of the data values are < or = x ex. If the 80th percentile is 28, then 80% of the data equal 28 or less

• Quartiles – the 25th, 50th, 75th percentiles. The 25th percentile is the lower or first quartile Q1, the 50th percentile is the median, the 75th percentile is the upper or third quartile Q3.

• Z-score – shows how many standard deviations a value is above or below the mean

How do I get the summary values?

• You can calculate most of the summary values using 1-Var Stats.

• The order on the calculator is:1-Var Stats L1 or 1-Var Stats L1, L2

The data values are in L1 and the frequencies are in L2

Categorical Data Displays

Frequency Tables

Grades Earned on Test 1Grade frequency A 10 B 15 C 5 D 2 F 1

Bar Chart

Segmented Bar Chart

Hobbies By Gender

Two Way TablesFavorite Leisure Activities

Dance Sports TV Total Men 2 10 8 20 Women 16 6 8 30 Total 18 16 16 50

One Other Graph – The Pie Chart

Sorry – couldn’t resistGOOD LUCK ON THE EXAM!!!