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Frequency Distribution Ibrahim Altubasi, PT, PhD The University of Jordan

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Statistical Notations SubjectsXYX-1XYX²X² 145 254 334 426 523 Σ Variable (s): X Y Variable (s): X Y Sample Size: n Summation: Σ ΣX = ΣX² = (ΣX)² = Σ(X-1) = ΣXY = Summation: Σ ΣX = ΣX² = (ΣX)² = Σ(X-1) = ΣXY =

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Descriptive Statistics Descriptive Statistics: statistical procedures used to summarize, organize, and simplify data. Descriptive Statistics: statistical procedures used to summarize, organize, and simplify data. Shape of Distribution Central Tendency Variability Frequency Distribution: Tables Graph Frequency Distribution: Tables Graph A display of the number (frequency) of individuals / observations in each value or category on the scale of measurement. A display of the number (frequency) of individuals / observations in each value or category on the scale of measurement.

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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

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Frequency Distribution Shape of the distribution Frequency distribution: Table Frequency distribution: Table Σf=n Calculate sum using frequency table : ΣX= Σ(f *X) Σf=n Calculate sum using frequency table : ΣX= Σ(f *X) Score (X)Frequency (f)F*X 10220 9545 8756 7321 6212 500 414 Σf=20ΣX=158

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Frequency Distribution Score (X) Frequency (f) F*XProportion (p) Percentag e (%) 10220.1010 9545.2525 8756.3535 7321.1515 6212.1010 50000 414.055 Σf=20ΣX=158Σp= 1Σper =100 Relative frequency Proportion: p=f / n Percentage: percentage = p*100 Relative frequency Proportion: p=f / n Percentage: percentage = p*100

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Frequency Distribution grouped frequency distribution : A frequency distribution where scores are grouped into intervals rather than listed as individual values. Used with a wide range of score values. grouped frequency distribution : A frequency distribution where scores are grouped into intervals rather 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. 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 table for continuous variable: 1) Number of groups (around 10-15 groups) 2) All class intervals should be the same width 3) 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) Rules for creating grouped frequency distribution table for continuous variable: 1) Number of groups (around 10-15 groups) 2) All class intervals should be the same width 3) 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)

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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

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Frequency Distribution Score (X)Frequency (f) 102 95 87 73 62 50 41 Σf=20 Class interval fReal limitMidpoint 4-513.5-5.54.5 6-755.5-7.56.5 8-9127.5-9.58.5 10-1129.5-11.510.5 Interval size =2 Score limit

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Frequency Distribution Shape of the distribution Frequency distribution: Table Graphs Histogram polygon Bar graph Frequency distribution: Table Graphs Histogram polygon Bar graph Interval and ratio scale data Nominal and ordinal scale data

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Frequency Distribution Shape of the distribution Frequency distribution: Table Graphs Histogram Frequency distribution: Table Graphs Histogram A graph showing a bar above each score or interval so that the height of the bar corresponds to the frequency and width extends to the real limits of the score or interval. Adjacent bars touch each other.

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Frequency Distribution Shape of the distribution Frequency distribution: Table Graphs Histogram polygon Bar graph Frequency distribution: Table Graphs Histogram polygon Bar graph A graph consisting of a line that connects a series of dots. A dot is placed above each score or interval so that the height of the dot corresponds to the frequency.

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Frequency Distribution Shape of the distribution Frequency distribution: Table Graphs Histogram polygon Bar graph Frequency distribution: Table Graphs Histogram polygon Bar graph A bar graph is the same as a histogram except that spaces are left between adjacent bars.

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Frequency Distribution Shape of the distribution Frequency distribution: Table Graphs Histogram polygon Bar graph Stem and leaf display Frequency distribution: Table Graphs Histogram polygon Bar graph Stem and leaf display Combines the characteristics of a graph and a table Score Stem-and-Leaf Plot FrequencyStem & Leaf 1.00 4. 0 2.00 6. 00 3.00 7. 000 7.00 8. 0000000 5.00 9. 00000 2.00 10. 00 Stem width: 1.00 Each leaf: 1 case(s)

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Frequency Distribution Stem Width: ? Each leaf: ? Cases

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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.

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Frequency Distribution Shape of a distribution Symmetric distribution Symmetrical distribution A distribution where the left-hand side is a mirror image of the right-hand side.

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Frequency Distribution Shape of a distribution Symmetric distribution Skewed distribution: positively skewed vs. negatively skewed A distribution where the scores pile up on the left side and taper off to the right. A distribution where the scores pile up on the right side and taper off to the left

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Frequency Distribution The cumulative frequency of a score value or class interval: the number of cases falling below the upper real limit of that score value or class interval. Cumulative percentage: the percentage of individuals with values at or below a particular point in the distribution. The cumulative percentage values are associated with the upper real limits of the corresponding scores or intervals.

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Frequency Distribution Score X Frequency f Cumulative frequency Proportion (p) Cumulative proportion Percentage %Cumulative % 10220.101.0010%100% 9518.25.9025%90% 8713.35.6535%65% 736.15.3015%30% 623.10.1510%15% 5010.050%5% 411.05 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)

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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. 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 X Frequency f Cumulative frequency Cumulative % 10220100% 951890% 871365% 73630% 62315% 5015% 411 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?

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Frequency Distribution Use of the Ogive to find percentiles and percentile ranks Ogive: Any continuous cumulative percentage curve.

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