frequency distribution ibrahim altubasi, pt, phd the university of jordan
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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.