Transcript 7.1 Visual Representations of Data

Statistics
Frequency Table
Graphical Representations
1.
2.
3.

4.
5.
6.
Bar Chart, Pie Chart, and Histogram
Median and Quartiles
Box Plots
Interquartile Range and Five-Number
Summary
1


Statistics is the branch of mathematics that
deals with data: their collection, description,
analysis, and use in prediction.
Data can be presented in raw form or
organized and displayed in tables or charts.
2

A table like the one below is called a frequency
table since it presents the frequency with which
each response occurs.
3

This graph shows the same data as the previous
example as a bar chart.
4

The pie chart consists of a circle subdivided into
sectors, where each sector corresponds to a
category. The area of each sector is proportional
to the percentage of items in that category. This
is accomplished by making the central angle of
each sector equal to 360 times the percentage
associated with the segment.
5

The pie chart of the data of the previous example
is:
6

When the data is numeric data, then it can be
represented by a histogram which is similar to a
bar chart but there is no space between the bars.
7
The grades for the first quiz in a class of 25
students are
8 7 6 10 5 10 7 1 8 0
10 5 9 3 8 6 10 4 9 10
7 0 9 5 8.
(a) Organize the data into a frequency table.
(b) Create a histogram for the data.
8
Grade
Number
10
5
9
3
8
4
7
3
6
2
5
3
4
1
3
1
2
0
1
1
0
2
9
10


The median of a set of numerical data is the data
point that divides the bottom 50% of the data
from the top 50%. To find the median of a set of
N numbers, first arrange the numbers in
increasing or decreasing order. The median is
the middle number if N is odd and the average
of the two middle numbers if N is even.
The quartiles are the medians of the sets of data
below and above the median.
11
For the grade data given,
(a) find the median;
(b) find the quartiles.
Grade
Number
10
5
9
3
8
4
7
3
6
2
5
3
4
1
3
1
2
0
1
1
0
2
12
N = 25 so median is the 13th grade: 7.
There are 12 grades in the lower and
upper halves.
The upper quartile is the average of the
19th and 20th grade:
Q3 = (9 + 9)/2 = 9.
The lower quartile is the average of the
6th and 7th grade:
Q1 = (5 + 5)/2 = 5.
Grade
Number
10
5
9
3
8
4
7
3
6
2
5
3
4
1
3
1
2
0
1
1
0
2
13

Graphing calculators can display a picture,
called a box plot, that analyzes a set of data and
shows not only the median, but also the
quartiles, lowest data point (min) and largest
data point (max).
14
For the grade data given,
find the box plot.
0
5
7
9
10
15
Grade
Number
10
5
9
3
8
4
7
3
6
2
5
3
4
1
3
1
2
0
1
1
0
2


The length of the rectangular part of the box
plot, which is Q3 - Q1, is called the interquartile
range.
The five pieces of information, min, max, Q2 =
median, Q1 and Q3 are called the five-number
summary.
16
For the grade data from
the previous example,
list the five-number
summary and the
interquartile range.
0
5
7
9
10
min = 0, Q1 = 5, Q2 = median = 7, Q3 = 9,
max = 10
Interquartile range is Q3 - Q1 = 9 - 5 = 4.
17


Bar charts, pie charts, histograms, and box plots
help us turn raw data into visual forms that
often allow us to see patterns in the data
quickly.
The median of an ordered list of data is a
number with the property that the same
number of data items lie above it as below it.
For an ordered list of N numbers, it is the
middle number when N is odd, and the
average of the two middle numbers when N is
even.
18

For an ordered list of data, the first quartile Q1
is the median of the list of data items below the
median, and the third quartile Q3 is the median
of the list of data items above the median. The
difference of the third and first quartiles is
called the interquartile range. The sequence of
numbers consisting of the lowest number, Q1,
the median, Q3, and the highest number is
called the five-number summary.
19