Statistical Graphs and Measures of Center in Sierra College Math 13 - Prof. John Burke, Study notes of Statistics

An overview of statistical graphics and measures of center in the context of sierra college math 13. It covers topics such as frequency polygons, ogives, dotplots, stem-and-leaf plots, and various measures of center including mean, median, mode, and midrange. The document also includes examples and explanations of how to calculate these measures.

Typology: Study notes

Pre 2010

Uploaded on 07/30/2009

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Sierra College – Math 13
Spring 2009 – Class 5/32
Today: Sections 2-4; 3-1/3-2
Assignment: 2-4 {1, 5, 7, 9, 11, 13, 15}
3-2 {1, 3, 5, 7, 9, 11, 13, 15, 17}
Next: Sections 3-3/3-4
Instructor: John Burke
Web Page: http://math.sierracollege.edu/Staff/JohnBur ke/
Telephone: 916 337-0425
Office hours: (V-307) MW 2:35-5:00; M 2:45-3:45 (official)
2
2-4 Statistical Graphics
The main objective in using graphic al representations
of data is to better understand the data through:
Description (center, shape, etc.),
Exploring (relative strengths), and
Comparing (data from different populations)
3
Frequency Polygon
A frequency polygon uses line segments connected
to points located directly above class midpoint
values. The heights of the points corr espond to the
class frequencies, and the line se gments are
extended to the right and left so that the graph begins
and ends on the horizontal axis.
13
10
7
4
1
112 – 14
29 – 11
156 – 8
143 – 5
200 – 2
Frequency
Rating
Class midpoints
pf3
pf4
pf5
pf8
pf9
pfa

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Sierra College – Math 13

Spring 2009 – Class 5/

Today: Sections 2-4; 3-1/3-

Assignment: 2-4 {1, 5, 7, 9, 11, 13, 15}

Next: Sections 3-3/3-

Instructor: John Burke

E-mail: [email protected]

Web Page: http://math.sierracollege.edu/Staff/JohnBurke/

Telephone: 916 337-

Office hours: (V-307) MW 2:35-5:00; M 2:45-3:45 (official)

2-4 Statistical Graphics

The main objective in using graphical representations

of data is to better understand the data through:

– Description (center, shape, etc.),

– Exploring (relative strengths), and

– Comparing (data from different populations)

Frequency Polygon

A frequency polygon uses line segments connected

to points located directly above class midpoint

values. The heights of the points correspond to the

class frequencies, and the line segments are

extended to the right and left so that the graph begins

and ends on the horizontal axis.

Rating Frequency

Class midpoints

Ogive

An ogive is a line graph that depicts the cumulative

frequencies, just as the cumulative frequency

distribution lists cumulative frequencies.

Frequency

Cumulative Frequency

Rating

Dotplot

A dotplot consists of a graph in which each data

value is plotted as a point (or dot) along a scale of

values. Dots representing equal values are stacked.

Stem-and-Leaf Plot

A stem-and-leaf plot represents data by separating

each value into two parts: the stem (such as the

leftmost digit(s)) and the leaf (such as the rightmost

digit). The stem-and-leaf plot shows the distribution,

but maintains all the information in the original list.

Raw Data (Test Grades)

Stem

Leaves

Time-Series Graph

A time-series graph is used to represent data values

collected over time.

Number of screens

at drive-in movie

theaters

Variation on the pie chart

developed by Florence Nightingale

Deaths in British Military

Hospitals During the

Crimean War

Deaths due to preventable disease

Deaths due to battle wounds

Deaths due to other

A representation of six different variables relevant to the

march of Napoleon’s army to Moscow and back in 1812-

developed in 1861 by Charles Joseph Minard.

3-2 Measures of Center

Definition: A measure of center is a value at the

center, or middle, of a data set.

We will consider the following measures of center

( the 4 M’s):

• Mean

• Median

• Mode

• Midrange

Mean

The arithmetic mean (or just mean or average ) of a

set of values is the measure of center found by adding

the values and dividing by the total number of values.

x

mean x

n

Note: The mean is particularly sensitive to outliers.

Called “x-bar”

Mean - Example

Mean = Σx/4 = 5.

Mean = Σx/5 = 9.

Notice the effect of the outlier.

Median - Example

No exact middle – even number of values

Median = (3.60 + 6.44)/2 = 5.

Mean = Σx/4 = 5.

Exact middle – median = 6.

Mean = Σx/5 = 9.

Notice the effect of the outlier.

Mode

The mode of a set of data, often denoted M, is the value

that occurs most frequently.

When two values occur with the same greatest frequency,

each is a mode and the data set is bimodal.

When more than two values occur with the same greatest

frequency, the data set is said to be multimodal.

When no value is repeated, there is no mode.

Mode is the only measure of central tendency that can be

used with nominal data.

Mode - Example

5, 5, 5, 3, 1, 5, 1, 4, 3, 5 Å Mode is 5

1, 2 ,2, 2, 3, 4, 5, 6, 6, 6, 7, 9 Å Bimodal – 2 and 6

1, 2, 3, 6, 7, 8, 9 Å No mode

Midrange

The midrange is the measure of center that is the value

midway between the highest and lowest values in the

original data set. It is found by taking the average (mean) of

the highest and lowest data values.

midrange = highest value + lowest value

Note: Rarely used because it is too sensitive to extreme

values.

Midrange - Example

Midrange = (6.72 + 3.46)/2 = 5.

Median = (3.60 + 6.44)/2 = 5.

Mean = Σx/4 = 5.

Midrange = (26.7 + 3.46)/2 = 15.

Exact middle – median = 6.

Mean = Σx/5 = 9.

Notice the effect of the outlier.

Calculating Measures of Center

Find the sum

of all values

then divide by

the number

of values

Sort the data Value that

occurs most

frequently

Max + Min

Mean Median Mode Midrange

Median is the

value in the

exact middle.

Add the two

middle numbers,

then divide by 2.

Even number

of values

Odd number

of values

Rarely used Good for nominal data

Good choice if there are extreme values

Sensitive to extreme values

Tends to vary less than other measures of center

Skewness

Data is symmetric if the left half of the histogram is

roughly a mirror image of the right half.

Data is skewed if it is not symmetric and extends

more to one side than the other.

Lopsided to the right = skewed to left = negatively skewed

Lopsided to the left = skewed right = positively skewed

Measures of Center Review

What are the four measures of center we looked at?

Mean, Median, Mode and Midrange

Which measure of center can be used with nominal data?

Mode