Frequency Distribution

Ibrahim Altubasi, PT, PhDThe University of Jordan

Statistical Notations Subject

sX Y X-

1XY

X²

1 4 5

2 5 4

3 3 4

4 2 6

5 2 3

Σ

Variable (s): X Y

Sample Size: n

Summation: ΣΣX =

ΣX² =

(ΣX)² =

Σ(X-1) =

ΣXY =

Descriptive Statistics Descriptive Statistics: statistical procedures used tosummarize, organize, and simplify data.

Shape of Distribution

Central Tendency

Variability

Frequency Distribution:

TablesGraph

A display of the number (frequency) ofindividuals / observations in each value orcategory on the scale of measurement.

Data• The following set of n=20 scores was obtained

from a 10-point statistics quiz:

8, 9, 8, 7, 10, 9, 6, 4, 9, 8, 7, 8, 10, 9, 8, 6, 9, 7, 8, 8

Frequency Distribution

Shape of the distribution

Frequency distribution:TableΣf=nCalculate sum usingfrequency table :ΣX= Σ(f *X)

Score (X)

Frequency (f)

F*X

10 2 20

9 5 45

8 7 56

7 3 21

6 2 12

5 0 0

4 1 4

Σf=20 ΣX=158

Frequency Distribution

Score (X)

Frequency (f)

F*X Proportion (p)

Percentage (%)

10 2 20 .10 10

9 5 45 .25 25

8 7 56 .35 35

7 3 21 .15 15

6 2 12 .10 10

5 0 0 0 0

4 1 4 .05 5

Σf=20 ΣX=158

Σp= 1 Σper =100

Relative frequencyProportion:p=f / nPercentage:percentage = p*100

Frequency Distribution

grouped frequency distribution :•A frequency distribution where scores are grouped into intervalsrather than listed as individual values.•Used with a wide range of score values.class interval:•A group of scores in a grouped frequency distribution•The size of a class interval is the number of score values within it.Rules for creating grouped frequency distribution tablefor continuous variable:1) Number of groups (around 10-15 groups)2) All class intervals should be the same width3) The width of each interval should be a relatively simple number (eg. 2, 3, 5, 10 and etc)4) The bottom score of each interval should be a multiple of the width(the lowest score value if it is divisible by the size,or the first number below the lowest score which is divisible by the interval size)

Frequency Distribution

Grouped Frequency Distribution for continuous variables

Score Limits: the score values that appear as the lowest and the highest scores in an interval

Lower real limit: the boundary that separates an interval from the next lower interval

Upper real limit: the boundary that separates an interval from the next upper interval

How to calculate the real limits: identify the point halfway between the upper score limit of a particular interval and the lower score limit of the next higher interval

How to calculate midpoint: divide the interval width in half; add this value to the lower real limit of the interval

Frequency Distribution

Score (X)

Frequency (f)

10 2

9 5

8 7

7 3

6 2

5 0

4 1

Σf=20

Class interva

l

f Real limit

Midpoint

4-5 1 3.5-5.5 4.5

6-7 5 5.5-7.5 6.5

8-9 12

7.5-9.5 8.5

10-11 2 9.5-11.5 10.5

Interval size =2

Score limit

Frequency Distribution

Shape of the distribution

Frequency distribution:TableGraphs Histogram polygon

Bar graph

Interval and ratio scale data

Nominal and ordinal scale data

Frequency Distribution

Shape of the distribution

Frequency distribution:TableGraphs Histogram

A graph showing a bar above each score orinterval so that the height of the barcorresponds to the frequency and widthextends to the real limits of the score orinterval. Adjacent bars touch each other.

Frequency Distribution

Shape of the distribution

Frequency distribution:TableGraphs Histogram polygon

Bar graph

A graph consisting of a line that connects aseries of dots. A dot is placed above each score or interval so that the height of the dot corresponds to the frequency.

Frequency Distribution

Shape of the distribution

Frequency distribution:TableGraphs Histogram polygon

Bar graph

A bar graph is the same as a histogram except that spaces are left between adjacent bars.

Frequency Distribution

Shape of the distribution

Frequency distribution:TableGraphs Histogram polygon Bar graph

Stem and leaf display

Combines the characteristics of a graph and a table

Score Stem-and-Leaf Plot

Frequency Stem & Leaf1.00 4. 02.00 6 . 003.00 7 . 0007.00 8 . 00000005.00 9 . 000002.00 10 . 00

Stem width: 1.00Each leaf: 1 case(s)

Frequency Distribution

Stem Width: ?Each leaf: ? Cases

Frequency Distribution

A grouped frequency distribution histogram and a stem and leaf display. The stem and leaf display is placed on its side to demonstrate that the display gives the same information provided in the histogram.

Frequency Distribution

Shape of a distribution

Symmetric distribution

Symmetrical distribution

A distribution where the left-handside is a mirror image ofthe right-hand side.

Frequency Distribution

Shape of a distributionSymmetric distribution

Skewed distribution: positively skewedvs. negatively skewed

A distribution where the scores pile up on the left side and taper off tothe right.

A distribution where thescores pile up on the right side and taper off to the left

Frequency Distribution

The cumulative frequency of a score value or class interval:the number of cases falling below the upper real limit ofthat score value or class interval.

Cumulative percentage:the percentage of individuals with values at or below a particularpoint in the distribution. The cumulative percentage values areassociated with the upper real limits of the corresponding scoresor intervals.

Frequency Distribution

Score X

Frequency f

Cumulative

frequency

Proportion (p)

Cumulative

proportion

Percentage %

Cumulative %

10 2 20 .10 1.00 10% 100%

9 5 18 .25 .90 25% 90%

8 7 13 .35 .65 35% 65%

7 3 6 .15 .30 15% 30%

6 2 3 .10 .15 10% 15%

5 0 1 0 .05 0% 5%

4 1 1 .05 .05 5% 5%

Among the 20 students, 6 had the score no more than 7 (7.5)Among the 20 students, 30% had the score no more than 7 (7.5)

Frequency Distribution

Percentiles and percentile ranks are used to describe the position of individual scores within a distribution.

Percentile rank of a score is defined as the percentage of individuals in the distribution with scores at or below the particular value.

Percentile is associated with a score.Score

XFrequency

fCumulativ

e frequency

Cumulative %

10 2 20 100%

9 5 18 90%

8 7 13 65%

7 3 6 30%

6 2 3 15%

5 0 1 5%

4 1 1 5%

The score 7 has a percentile rank of 30%.

The 30th percentile is 7 (7.5).

What is the 15th percentile?What is the percentile rank of 9?

Frequency Distribution

Use of the Ogive to find percentiles and percentile ranks

Ogive: Anycontinuouscumulativepercentage curve.