Quartiles Q1, Q2, Q3 of the series of distributing class intervals - calculator
Quartiles Q1, Q2, Q3 of the interval series
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Quartiles information
Quartile is one of the measures of central tendency used to determine the value of the feature around which the data is grouped. Therefore, it is about concentrating most of the data around some representative of the examined feature. First (lower) quartile Q1 divides the group ordered into two parts in such a way that 25% of the units have a value not greater than Q1 , and the remaining 75% equal or greater than this quartile; Formula for the estimate of the first quartile:
nQ1 - frequency of the first quartile (lower quartile)0Q1 - lower class boundary of the range containing the first quartile isk-1 - cumulative frequency range before the cumulative frequency of the first quartileQ1 = x1Q1(higher) - x0Q1(lower) - size (width) of the range containing the first quartileQ1 = \(\frac{n}{4}\) - first quartile number (position) for even observationsQ1 = \(\frac{n+1}{4}\) - first quartile number (position) for odd observations Second (middle) quartile Q 2 (median, middle value) divides the ordered group into two parts in such a way that half of the group's units have values of the variable equal to or greater than the median hence the median is sometimes called the middle value. Formula for the estimate of the second quartile:
nQ2 - frequency of the second quartile (middle quartile)0Q2 - lower class boundary of the range containing the second quartileisk-1 - cumulative frequency range before the cumulative frequency of the second quartileQ2 = x1Q2(higher) - x0Q2(lower) - size (width) of the range containing the second quartileQ2 = \(\frac{n}{2}\) - second quartile number (position) for even observationsQ2 = \(\frac{n+1}{2}\) - second quartile number (position) for odd observations The third (upper) quartile Q3 divides the ordered group in such a way that 75% of units have feature values not higher than Q3 and the remaining 25 % not lower than Q3 .
Formula for the estimate of the third quartile:
nQ3 - frequency of the third quartile (upper quartile)0Q3 - lower class boundary of the range containing the third quartileisk-1 - cumulative frequency range before the cumulative frequency of the third quartileQ3 = x1Q3(higher) - x0Q3(lower) - size (width) of the range containing the third quartileQ3 = \(\frac{3·n}{4}\) - third quartile number (position) for even observationsQ3 = \(\frac{3·(n+1)}{2}\) - third quartile number (position) for odd observations
EXAMPLE Distance of residence in km. Number of employees 0-5 5 5-10 25 10-15 30 15-20 55 20-25 30 25-30 20 30-35 15
Number of employees column one by one. The calculations are presented in the table below.Distance of residence in km. Number of employees Cumulative frequency 0-5 5 0+5=5 5-10 25 5+25=30 10-15 30 30+30=60 15-20 55 60+55=115 20-25 30 115+30=145 25-30 20 145+20=165 30-35 15 165+15=180
1 ,Q2 ,Q3 .1 \(N_{Q_1}=\frac{n}{4}\) 2 \(N_{Q_2}=\frac{n}{2}\) 3 \(N_{Q_3}=\frac{3·n}{4}\) n this is the size of surveyed population. In our example, it is the number of all employees, n = 180, i.e. even (that's why we used the formula for even observations). Substituting to the formulas we get: cumulative frequencies column for the first value greater or equal to the quartile position. This will be our task:Q1 the value is 60 . Because 60 is the first cumulative frequency greater than the number (position) of the 1st quartile - 45 . So we can see that Q1 belongs to the range 10-15 km , whose number (number of employees) is 30 .Q2 the value is 115 . Because 115 is the first cumulative frequency greater than the number (position) of the second quartile - 90 . So we can see that Q2 belongs to the range 15-20 km , whose number (number of employees) is 55 .Q3 the value is 145 . Because 145 is the first cumulative frequency greater than the number (position) of the third quartile - 135 . So we can see that Q3 belongs to the range 20-35 km , whose number (number of employees) is 30 .Quartile Q1 x0Q1 = 10 - lower class boundary of the range containing the first quartile. Earlier, we established that this interval is 10-15 km, i.e. its lower limit will be 10.NQ1 = 45 - position (number) of the first quartile, which we calculated earlier from the formula \( N_{Q1}=\frac{n}{4} \)nisk-1 = 30 - cumulative frequency of the range that precedes the cumulative frequency to which the first quartile belongs. We determined that the cumulative frequency to which the first quartile belongs is 60. Looking at the row above in the table, we can see that the preceding cumulative number is 30.hQ1 = x1Q1(higher) - x0Q1(lower) = 15-10 = 5 - size (width) of the range containing the first quartile. As we already know, the range of the first quartile is 10-15 km, so to calculate the width from the upper value, we subtract the lower value.nQ1 = 30 - frequency of the range (number of employees) containing the first quartile.Quartile Q2 x0Q2 = 15 - lower class boundary of the range containing the second quartile. Earlier, we established that this interval is 15-20 km, i.e. its lower limit will be 15.NQ2 = 90 - position (number) of the second quartile, which we calculated earlier from the formula \( N_{Q2}=\frac{n}{2} \)nisk-1 = 60 - cumulative frequency of the range that precedes the cumulative frequency to which the second quartile belongs. We determined that the cumulative frequency to which the second quartile belongs is 115. Looking at the row above in the table, we can see that the preceding cumulative number is 60.hQ2 = x1Q2(higher) - x0Q2(lower) = 20-15 = 5 - size (width) of the range containing the second quartile. As we already know, the range of the second quartile is 15-20 km, so to calculate the width from the upper value, we subtract the lower value.nQ2 = 55 - frequency of the range (number of employees) containing the second quartile.Quartile Q3 x0Q3 = 20 - lower class boundary of the range containing the third quartile. Earlier, we established that this interval is 20-25 km, i.e. its lower limit will be 20.NQ3 = 135 - position (number) of the third quartile, which we calculated earlier from the formula \( N_{Q3}=\frac{3·n}{2} \)nisk-1 = 115 - cumulative frequency of the range that precedes the cumulative frequency to which the third quartile belongs. We determined that the cumulative frequency to which the third quartile belongs is 145. Looking at the row above in the table, we can see that the preceding cumulative number is 115.hQ3 = x1Q3(higher) - x0Q3(lower) = 25-20 = 5 - size (width) of the range containing the third quartile. As we already know, the range of the third quartile is 20-25 km, so to calculate the width from the upper value, we subtract the lower value.nQ3 = 30 - frequency of the range (number of employees) containing the third quartile.
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