Organizing Data Data when collected in original form is called “raw data”. For example.

Organizing DataData when collected in original form is called

“raw data”.For example

Frequency DistributionWe organize the raw data by using a

frequency distribution.The frequency is the number of values in a

specific class of data.A frequency distribution is the organizing of

raw data in table form, using classes and frequencies.

Frequency Distribution Cont.For the raw data set, a frequency distribution

is shown as follow:

We use a frequency distribution to construct a histogram.

Class limits Tally Frequency

1-3 ///// ///// 10

4-6 ///// ///// //// 14

7-9 ///// ///// 10

10-12 ///// / 6

13-15 ///// 5

16-18 ///// 5

HistogramsHistograms are bar graphs in whichThe bars have the same width and always

touch (the edges of the bars are on class boundaries which are described below).

The width of a bar represents a quantity.The height of each bar indicates

frequency.

To construct a histogram from raw data:Decide on the number of classes (5 to 15 is

customary).Find a convenient class width.Organize the data into a frequency table.Find the class midpoints and the class

boundaries.Sketch the histogram.

To find Class WidthFirst compute: Largest value - smallest

value Desired number of classes

Increase the value computed to the next highest whole number.

The lower class limit of a class is the lowest data that can fit into the class, the upper class limit is the highest data value that can fit into the class. The class width is the difference between lower class limits of adjacent classes.

Class WidthRaw Data:10.2 18.7 22.3 20.06.3 17.8 17.1 5.02.4 7.9 0.3 2.5 8.5 12.5 21.4 16.5 0.4 5.2 4.1 14.3 19.5 22.5 0.0 24.711.4

Use 5 classes.24.7 – 0.0 5= 4.94

Round class width up to 5.

Frequency TableDetermine class width.Create the classes. May use smallest data

value as lower limit of first class and add width to get lower limit of next class.

Tally data into classes.Compute midpoints for each class.Determine class boundaries.

Tallying the Data

tally frequency

0.0 - 4.9 |||| | 6

5.0 - 9.9 |||| 5

10.0 - 14.9 |||| 4

15.0 - 19.9 |||| 5

20.0 - 24.9 |||| 5

Computing Class Width

difference between the lower class limit of one class and the lower class limit of the next class

f class widths

0.0 - 4.9 6 5

5.0 - 9.9 6 5

10.0 - 14.9 4 5

15.0 - 19.9 5 5

20.0 - 24.9 5 5

Finding Class Widths

Computing Class Midpoints

lower class limit + upper class limit 2

f class midpoints

0.0 - 4.9 6 2.45

5.0 - 9.9 5 7.45

10.0 - 14.9 4 12.45

15.0 - 19.9 5 17.45

20.0 - 24.9 5 22.45

Finding Class Midpoints

Class BoundariesUpper limit of one class + lower limit of next class

2Class boundaries cannot belong to any class. Class boundaries between adjacent classes

are the midpoint between the upper limit of the first class, and the lower limit of the higher class.

Differences between upper and lower boundaries of a given class is the class width.

f class boundaries

0.0 - 4.9 6 0.05 - 4.95 5.0 - 9.9 5 4.95 - 9.95

10.0 - 14.9 4 9.95 - 14.95

15.0 - 19.9 5 14.95 - 19.95

20.0 - 24.9 5 19.95 - 24.95

Finding Class Boundaries

f

0.0 - 4.9 6

5.0 - 9.9 5

10.0 - 14.9 4

15.0 - 19.9 5

20.0 - 24.9 5

Constructing the Histogram f

| | | | | |

6

5

4

3

2

1

0

-

-

-

-

-

-

-

-0.05 4.95 9.95 14.95 19.95 24.95

Relative FrequenciesThe relative frequency of a class is f/n where

f is the frequency of the class, and n is the total of all frequencies.

Relative frequency tables are like frequency tables except the relative frequency is given.

Relative frequency histograms are like frequency histograms except the height of the bars represent relative frequencies.

f relative frequency

0.0 - 4.9 6 0.24

5.0 - 9.9 5 0.20

10.0 - 14.9 4 0.16

15.0 - 19.9 5 0.20

20.0 - 24.9 5 0.20

Relative Frequency Histogram

| | | | | |

.24

.20

.16

.12

.08

.04

0

-

-

-

-

-

-

-

-0.05 4.95 9.95 14.95 19.95 24.95

Rela

tive f

requ

ency

f/n

Common Shapes of HistogramsCommon Shapes of Histograms

Symmetrical

ff

When folded vertically, both sides are (more or less) the same.

Common Shapes of HistogramsCommon Shapes of Histograms

Also Symmetrical

ff

Uniform

ff

Common Shapes of HistogramsCommon Shapes of Histograms

Skewed Histograms

Skewed left Skewed right

Common Shapes of HistogramsCommon Shapes of Histograms

Bimodal

ff

The two largest rectangles are approximately equal in height and are separated by at least one class.

Frequency PolygonA frequency polygon

or line graph emphasizes the continuous rise or fall of the frequencies.

Dots are placed over the midpoints of each class.

Dots are joined by line segments.

Zero frequency classes are included at each end.

f

2 - 4 6

5 - 7 5

8 - 10 4

11 - 13 5

Constructing the Frequency Polygon

f

| | | | | |

6

5

4

3

2

1

0

-

-

-

-

-

-

-

0 3 6 9 12 15

Cumulative Frequencies & OgivesThe cumulative frequency of a class

is the frequency of the class plus the frequencies for all previous classes.

An ogive is a cumulative frequency polygon.

f

Greater than 1.5 20

Greater than 4.5 14

Greater than 7.5 9

Greater than 10.5 5

Greater than 13.5 0

Cumulative Frequency Table

f

2 - 4 6

5 - 7 5

8 - 10 4

11 - 13 5 20

f

Greater than 1.5 20

Greater than 4.5 14

Greater than 7.5 9

Greater than 10.5 5

Greater than 13.5 0

Constructing the Ogive

Cu

mu

lati

ve f

req

uen

cy

| | | | | |

20

15

10

5

0

-

-

-

-

-

1.5 4.5 7.5 10.5 13.5 pounds