STATISTICS

CHAPTER 5: FractilesKyna B. DavidFRACTILES: INTRODUCTION Fractiles are measures of location or position which include not only central location but also any position based on the number of equal divisions in a given distribution. If we divide the distribution into four equal divisions, then we have quartiles denoted by Q1, Q2, Q3, and Q4. The most commonly used fractiles are the quartiles, deciles, and percentiles.FRACTILES: INTRODUCTIONFractiles divide a data set into consecutive intervals so that each interval has (at least approximately) the same number of data values.Fractilesare data values that divide an ordered set into equal parts.

FRACTILES FOR UNGROUPED DATAQuartiles divide a distribution into four equal parts. For example, Q1, or the first quartile, locates the point which is greater than 25% of the items in distribution.In descriptive statistics, the quartiles of a set of values are the three points that divide the data set into four equal groups, each representing a fourth of the population being sampled.QuartilesFRACTILES FOR UNGROUPED DATAThe term quartile is derived from the word quarter which means one fourth of something. Thus a quartile is a certain fourth of a data set. When you arrange a date set increasing order from the lowest to the highest, then you divide this data into groups of four, you end up with quartiles.Quartilesdivide the set into four parts. About one quarter of the data falls on or below the first quartile, half below the second quartile (the median) and three fourths below the third quartile.QuartilesFRACTILES FOR UNGROUPED DATA Q3 is the 3rd quartile Q3 = 3N th item 4This means that 75% of the observations lie below this value.QuartilesFRACTILES FOR UNGROUPED DATAQ2 is the 2nd quartile Q2 = 2N th item or the median 4

This means that 50% of the observations lie below this value.QuartilesFRACTILES FOR UNGROUPED DATAQ1 is the 1st quartile Q1 = N th item 4

This means that 25% of the observations lie below this value.QuartilesFRACTILES FOR UNGROUPED DATADeciles are values that divide a distribution into 10 equal parts.Deciles refers to one of ten equal groups which are divided a large group of values or statistics.It is any one of the numbers or values in a series dividing the distribution of the individuals in the series into ten groups of equal frequency.DecilesFRACTILES FOR UNGROUPED DATAD1 is the 1st decile. D1 = N th item 10D3 is the 3rd decile. D3 = 3N th item 10D5 is the 5th decile. D5 = 5N th item or the median 10DecilesFRACTILES FOR UNGROUPED DATAPercentiles are values that divide the distribution into 100 equal parts. P10 or the tenth percentile means the 10th item in the distribution which is 10% higher than the rest of the items.In statistics, a percentile (or centile) is the value of a variable below which a certain percent of observations fall.PercentilesFRACTILES FOR UNGROUPED DATAA percentile is a certain percentage of a set of data. Percentiles are used to observe how many of a given set of data fall within a certain percentage range.Percentilesdivide the data into 100 parts.A measure that tells us what percent of the total frequency scored at or below that measuremost often used for determining the relative standing of an individual in a population or the rank position of the individual.PercentilesFRACTILES FOR UNGROUPED DATAP1 is the 1st percentile P1 = Nth item 100 P25 is the 25th percentile P25 = 25Nth item or Q1 100P50 is the 50th percentile P50 = 50Nth item or the median 100

P67 is the 67th percentile P67 = 67Nth item 100PercentilesFRACTILES FOR UNGROUPED DATACalculate Q1, Q2, Q3, D1, D4, D5, D7, P10, P25, P50 and P70 for the following IQ scores: 87 90 95 96 97 98 98 99 100 100 100 100 100 101 101 102 102 102 103 104 105 107 110

Q1 = N th = 23 th = 5.75th item, which is 98. 4 4ExamplesFRACTILES FOR UNGROUPED DATANote: For odd number of observations, when the item number is fractional, take the next higher item. Since the 5th item 97, the 5.75th item is the next value, which is 98. This means that the score of 98 is higher than 25% of the items in the distribution. If the number of cases is even, take the point midway between the two items located at the middle of the distribution.ExamplesFRACTILES FOR UNGROUPED DATA Q2 = 2N th = 2(23) th = 23 th = 11.5th item, 4 4 2 which is 100. This means that the score of 100 is higher than 50% of the items in distribution. Q3 = 3N th = 3(23) th = 69 th = 17.25 th item, 4 4 4which is 102. D1 = N th = 23 th = 2.3 th item, which is 95. 10 10 ExamplesFRACTILES FOR UNGROUPED DATA Q2 = 2N th = 2(23) th = 23 th = 11.5th item, 4 4 2 which is 100. This means that the score of 100 is higher than 50% of the items in distribution. Q3 = 3N th = 3(23) th = 69 th = 17.25 th item, 4 4 4which is 102. D1 = N th = 23 th = 2.3 th item, which is 95. 10 10 ExamplesFRACTILES FOR UNGROUPED DATAD4 = 4N th= 4(23)th= 92 th=9.2th item, which is 100. 10 10 10D5= 5N th= 5(23)th= 115 th= 11.5th item, which is 100. 10 10 10D7= 7N th=7(23)th= 161 th=16.1th item, which is 102. 10 10 10P10= 10N th=10(23)th= 23 th=2.3th item, which is 95. 100 100 10 P25= 25N th=25(23)th= 23 th=5.7th item, which is 98. 100 100 4

ExamplesFRACTILES FOR UNGROUPED DATAP50= 50Nth=50(23)th= 23th=11.5th item, which is 100. 100 100 2P70= 70Nth=70(23)th= 1,610th=16.1th item, which is 102 100 100

NOTE: that median is equal to Q2, D5, and P50ExamplesFRACTILES FOR UNGROUPED DATAEstimate the cumulative frequencies on the tableFind N/4 or one-fourth of the number of the cases in the distribution.Determine the class limit in which N/4 case fallsCompute Q1 by using the formula; Q1 = L+C (N/4-Cf