Chapter FourMaking and Describing Graphs of Quantitative Variables

Center, Shape. and Spread

Describing Data With Graphs

Histograms, Stem and Leaf Plots, Dot Plots

Histograms

Histograms are bar charts for QUANTITATVE data where the upper class limit of one class is the same as the lower class limit of the next class. As a result, the bars are next to each other in a histogram.

Dem: Histogram

Stem and Leaf Plots

In a stem and leaf plot, data is sorted to reveal the underlying distribution. To construct a stem and leaf plot:

1) Separate the right most digit from the rest of the digits. This right most digit becomes the leaf.

2) Insert a legend into the stem and leaf plot indicating the position of this leaf.

3) The rest of the number without the leaf becomes the stem.

4) Organize the stems in order, draw a vertical line and place the leaves in order on the other side of the vertical line.

Stem and Leaf Plot Example

For the data: 25.6, 25.2, 25.0, 24.1, 24.9, 23.8, 23.6, 23.6, 25.7, 22.0 Organize a stem and leaf plot.

Applet for Stem and Leaf Graph

Dot Plots

A dot plot consists of a graph in which each data value is plotted as a point (dot) along a scale of values. When values occur more than once, they are plotted as dots stacked vertically above the scale value.

Dot Plot Example

Give a dot plot for the following values:

201, 201, 200, 203, 203, 203, 205, 205,207

Outliers

Outlier – an observation that is unusually large or small relative to the other values in the data set.

Causes of Outliersa) the measurement is incorrectly observed, recorded, or entered into a computerb) the measurement comes from a different population c) the measurement is correct but represents a rare event

Numerical Measures

Another way to give information about a sample or a population besides a graph is to use numerical measures about the sample or population.

Numerical measures about a sample are called statistics.

Numerical measures about a population are called parameters.

Numerical Measures of Central Tendency

MeanMode MedianMidrange

Mean

Arithmetic Mean – the sum of the measurements divided by the total number of measurements.

Population symbol for mean is µ

( µ is a parameter)

The symbol for the mean of a sample is x bar ( x bar is a statistic)

n

xxxmean n

...21

__

x

Finding the Arithmetic Mean from a Frequency Table

Class Freq (f)

Class

Mark (x)

200<210

6 205 1230

210<220

10 215 2150

220<230

14 225 3150

Total 30 6530

Mean 6530/30=217.7

xf

Round-Off Rule

Carry one more decimal place than is present in the original data set. DO NOT ROUND UNTIL THE END. Let your calculator carry as many decimal places as it can until the very end.

Other Measures of Central Tendency

Mode – the most popular ( the data item that occurs the most often)

Median – when data is ranked, the median is the middle piece of data. When there is an even amount of data, the two middle pieces of data are added together and divided by two.

Midrange – the highest plus lowest data entries divided by two. (We will not use this.)

Shape of Graphs

Symmetric Approximately symmetric Uniform (or approximately uniform) Skewed right Skewed left Uni-modal Bi-modal Gap Data Far from others How Class Width Affects a Histogram

Skewness

A distribution of data is SKEWED if it is not symmetric and extends more to one side than the other. (A distribution is symmetric if the left half of its histogram is roughly a mirror image of its right half.)

Skewness

1) Skewed LEFT (Negatively skewed): The mean and the median are to the left of the mode.

2) Symmetric (Zero Skewness): The mean, median, and the mode are the same.

3) Skewed right (Positively Skewed): The mean and the median are to the right of the mode.

Demo of Skewness and the Mean

Spread

Spread measures the variation of the data.(How far apart is the lowest data point from the

highest data point?) Three measures of spread are:

Standard deviation Variance

Range

We will talk more about spread later.

Women versus Men – Study Time

For the followeing data sets, prepare a histogram, dot-plot, and stem and leaf plot. Compare center, shape, and spread for the two sets of data.