FREQUENCY DISTRIBUTION TABLES

FREQUENCY DISTRIBUTION GRAPHS

Frequency Distribution: Lists each category (label) of data and the number of occurrences.

Sum of all = population or sample size

Relative Frequency: The proportion of occurrences for each category calculated as:

Sum of all = 1.

_ _ _

Frequency

Sum of all frequencies

Bar Graph: Vertical or Horizontal. X-axis contains the categories or labels. For Frequency Distributions the y-axis is the number of occurrances. For Relative Frequency Distributions the y-axis is the proportion (values between 0 and 1). Bars do not need to be touching.

SPECIES ZOO A ZOO B

Elephant 6 3

Giraffe 12 7

Impala 13 24

Zebra 1 2

Ostrich 6 1

Guinea Hens 25 12

ZOO A ANIMAL INVENTORY

05

1015202530

Elepha

nt

Giraffe

Impala

Zebra

Ostrich

Guinea

Hen

s

SPECIES

NU

MB

ER

ZOO A ANIMAL INVENTORY

05

1015202530

Elepha

nt

Giraffe

Impala

Zebra

Ostrich

Guinea

Hen

s

SPECIES

NU

MB

ER

ZOO B ANIMAL INVENTORY

05

1015202530

Elepha

nt

Giraffe

Impala

Zebra

Ostrich

Guinea

Hen

s

SPECIES

NU

MB

ER

ZOO A ANIMAL INVENTORY

05

1015202530

Guinea

Hen

s

Impala

Giraffe

Elepha

nt

Ostrich

Zebra

SPECIES

NU

MB

ER

CAN TREAT DISCRETE DATA LIKE QUALITATIVE (IF ONLY SEVERAL VALUES) OR AS WE WILL BE TREATING CONTINUOUS DATA (IF MANY VALUES)

SEPARATE CONTINUOUS DATA INTO CLASSES (INTERVALS) AND THEN DO DISTRIBUTION TABLES OR GRAPHS

Frequency Distribution Table: Similar to that for qualitative data, but each class is for a value or an interval (range) of values.

Histograms: Vertical bar graphs, where the x-axis is the number line and each bar is for a class. All bars must touch side to side. Uses Lower Class limit on x-axis.

Cumulative Frequency Distributions: Each class listed as before (lowest to largest), but the frequencies are the total for that frequency and all the lower classes.

Relative Cumulative Frequency Distribution: Each Cumulative Frequency divided by total of all frequencies. The last class will have a cumulative value of 1.0

Use number of siblings Do as Frequency Table Do as Relative Frequency Do as Cumulative Frequency Do as Relative Cumulative Frequency

Class: An interval of numbers along the number line.

Lower Class Limit (LCL): The beginning number of the class.

Upper Class Limit (UCL): The last number of the class.

Class Width: the difference between lower class limits (or upper class limits), found by taking using data set’s maximum and minimum and calculating

rounding up to a convenient value

Midpoint of Each Class: The point in the middle of the class, found by averaging the class lower class limit and the next class lower class limit.

#_ _

Maximum Minimum

of Classes

1. Organize data in ascending order:

1.03 1.72 1.99 3.21 4.24 4.58

1.36 1.75 2.52 3.47 4.27 4.72

1.45 1.85 2.67 3.50 4.43 4.75

1.51 1.92 3.06 3.72 4.54 4.79

1.63 1.95 3.20 3.78 4.57 4.91

2. Determine the number of classes (5 – 20): For this we will use 6.

3. Find the maximum and minimum: For this max = 4.91 and min = 1.03

4. Calculate the Class Width:

Round UP to a convenient value. We will use 0.70.

4.91 1.03 0.6476#_ _

MAX MINof CLASSES

5. Determine First Lower Class Limit: For this we will use 1.00 (something convenient and lower than the Minimum).

6. Determine the next 5 Lower Class Limits by adding class width to the first and each subsequent to get the next:

1.00+.70=1.70; 1.70+.70=2.40 … 3.10, 3.80, 4.50.

7. Determine the first Upper Class Limit by Subtracting 1 from the last place of the second Lower Class Limit: 1.70-.01=1.69.

8. Find the other 5 Upper Class Limits by adding the class width to each previous Upper Class Limits: 1.69+.7=2.39,

2.39+.7=3.09, …, 3.79, 4.49, 5.19

9. Now construct the Table ……:

CLASSLOWER CLASS

LIMITUPPER CLASS

LIMITFREQUENC

Y

1.00 1.69 ?

1.70 2.39 ?

2.40 3.09 ?

3.10 3.79 ?

3.80 4.49 ?

4.50 5.19 ?

1.03 1.72 1.99 3.21 4.24 4.58

1.36 1.75 2.52 3.47 4.27 4.72

1.45 1.85 2.67 3.50 4.43 4.75

1.51 1.92 3.06 3.72 4.54 4.79

1.63 1.95 3.20 3.78 4.57 4.91

And count the frequencies in each class …:

And complete the Table:

CLASSLOWER CLASS

LIMITUPPER CLASS

LIMITFREQUENC

Y

1.00 1.69 5

1.70 2.39 6

2.40 3.09 3

3.10 3.79 6

3.80 4.49 3

4.50 5.19 5

10. Draw the histogram:

5.3 5.8 6.4 7.1

5.5 5.9 6.6 7.1

5.6 6.2 6.6 7.3

5.7 6.3 6.7 7.6

5.7 6.3 6.8 7.9

Stem Leaf Plot: Used for recording and showing dispersion of data. Stem can be the integer portion of a number and the leaves the decimal portion. Or the stem could be the tens digit and the leaves the ones digit.

5-3,5,6,7,7,8,9 6-2,3,3,4,6,6,7,8 7-1,1,3,6,9

Dot Plot: Also used to show dispersion of data. Draw a number line and label the horizontal scale with the numbers from the data from lowest to highest. Then place a dot above the numbers each time the number occurs.

* * * * * * * * * * |___|___|___|___|___|___|___|___|___|___| 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3

Polygon Plot: Line graph using the midpoints for the x-axis and frequencies for the y-axis. Both ends of the line must come back to the 0 on the y-axis.

POLYGON GRAPH

0

2

4

6

8

10

0 2 4 6 8 10 12 14 16 18 20 22 24

CLASS MIDPOINT

FR

EQ

UE

NC

Y

Given a Polygon Plot, construct a Frequency Distribution Table.

› 1. Find the Class Width: Difference in Midpoints

› 2. Find first two LCL’s: Midpoint +/- ½*Class Width

› 3. Find First Upper Class Limit: 2nd LCL – 1› Find remainder of LCL’s & UCL’s› Find each class’s frequency

Ogive (pronounced oh jive) Plot: Line Graph used for displaying Cumulative Frequency Distributions. The x-axis is the Upper Class Limit and the y-axis is the Cumulative Frequency. The first point is a class width less than the first Upper Class Limit so that the line starts with a frequency of 0.

Ogive Plot:

Time Series Plots: Can be vertical or horizontal bar graphs, or line graphs. X-axis is time intervals or ages (years, months, days) and y-axis is frequency.

NORMAL DISTRIBUTION

0

2

4

6

8

10

A B C D F

GRADE

FR

EQ

UE

NC

Y

UNIFORM DISTRIBUTION

0

1

2

3

4

5

6

7

A B C D F

GRADE

FR

EQ

UE

NC

Y

SKEWED RIGHT DISTRIBUTE

0

2

4

6

8

10

12

A B C D F

GRADE

FREQ

UENC

Y

SKEWED LEFT DISTRIBUTE

0

2

4

6

8

10

12

A B C D F

GRADEFR

EQUE

NCY

Vertical Scale Manipulation: Not starting the y-axis at 0. Also using a break in the scale. Can make differences look bigger than they really are.

Exaggeration of Bars or Symbols: Used in pictographs.

Horizontal Scale Manipulation: Not all classes or time interval are the same width.

“Get your facts first, then you can distort then as you please” Mark Twain

“There are lies, damn lies, and STATISTICS” Mark Twain

“Definition of Statistics: The science of producing unreliable facts from reliable figures.” Evan Esar