Mathematics
Quartiles
Quartiles are values that divide a data set into four equal parts, each representing 25% of the data. The first quartile (Q1) marks the 25th percentile, the second quartile (Q2) is the median, and the third quartile (Q3) represents the 75th percentile. Quartiles are useful for understanding the spread and distribution of a dataset.
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12 Key excerpts on "Quartiles"
eBook - PDF- Ken Black, Ignacio Castillo, Amy Goldlist, Timothy Edmunds(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
Quartiles Quartiles are quantiles that divide a data set into four parts. The three Quartiles are denoted as Q 1 , Q 2 , and Q 3 . The first quartile, Q 1 , separates the first, or lowest, quarter of the data from the up- per three quarters and is equal to the 25th percentile. The second quartile, Q 2 , separates the bot- tom half of the data from the top half. Q 2 is located at the 50th percentile and equals the median of the data. The third quartile, Q 3 , divides the first three quarters of the data from the last quarter and is equal to the value of the 75th percentile. These three Quartiles are shown in Figure 3.6. FIGURE 3.5 Excel’s PERCENTILE Functions A D E C 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 B 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 #NUM! #NUM! #NUM! 7.45 10.6 12.25 12.7 13.15 13.6 14.15 15.5 16.85 17.8 18.7 20.2 22 24 26.25 #NUM! #NUM! 5 7.45 9.9 12.05 12.4 12.75 13.1 13.45 13.8 14.45 15.5 16.55 17.4 18.1 18.8 20 21.4 22.8 24.5 26.25 Dataset Desired Percentile PERCENTILE.EXC() PERCENTILE.INC() 14 12 19 23 5 13 28 17 Q 1 Q 2 Q 3 1st one quarter 1st two quarters 1st three quarters FIGURE 3.6 Quartiles As with percentiles (and other quantiles), there is universal agreement that the three quar- tiles divide a data set into four groups, but there are several different definitions for exactly how the Quartiles should be calculated. Just like with percentiles, there are a number of methods for dealing with situations when the size of the data set does not allow it to be divided into four groups with exactly the same number of data points. The method for quartile calculation that is described here is not the only widely accepted method, but it has the value of being consistent with the percentile calculation method that was described earlier.
No longer available |Learn more- William Mendenhall, Robert Beaver, Barbara Beaver, , William Mendenhall, Robert Beaver, Barbara Beaver(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
Calculate the quartile by finding a value either one-fourth, one-half, or three-fourths of the way between these two measurements. Need to Know… ? Many of the numerical measures that you have learned are easily found using com-puter programs or graphics calculators. The MINITAB command Stat ➤ Basic Statistics ➤ Display Descriptive Statistics , the Excel command Data ➤ Data Analysis ➤ Descrip-tive Statistics and the TI-83/84 Plus command stat ➤ CALC ➤ 1 -VAR STATS (see the Technology Today section at the end of this chapter) produce output containing the mean, the standard deviation, the median, the maximum and minimum, as well as the values of other statistics that we have not discussed. The data from Example 2.15 produced the MINITAB output shown in Figure 2.15. Notice that the Quartiles are identical to the hand-calculated values in that example. ■ The Five-Number Summary and the Box Plot The median and the upper and lower Quartiles shown in Figure 2.14 divide the data into four sets, each containing an equal number of measurements. If we add the largest number (Max) and the smallest number (Min) in the data set to this group, we will have a set of numbers that provide a quick and rough summary of the data distribution. The five-number summary consists of the following numerical measures: Q Q Min Median Max 1 3 By definition, one-fourth of the measurements in the data set lie between each of the four adjacent pairs of numbers. Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3 Maximum x 10 0 13.50 1.98 6.28 4.00 8.75 12.00 18.50 25.00 Descriptive Statistics: x Statistics Figure 2.15 MINITAB output for the data in Example 2.15 Copyright 2020 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience.
eBook - PDF- Prem S. Mann(Author)
- 2016(Publication Date)
- Wiley(Publisher)
These three measures are the first quar- tile (denoted by Q 1 ), the second quartile (denoted by Q 2 ), and the third quartile (denoted by Q 3 ). The data should be ranked in increasing order before the Quartiles are determined. The quar- tiles are defined as follows. Note that Q 1 and Q 3 are also called the lower and the upper Quartiles, respectively. Quartiles Quartiles are three values that divide a ranked data set into four equal parts. The second quartile is the same as the median of a data set. The first quartile is the median of the observations that are less than the median, and the third quartile is the median of the observations that are greater than the median. Figure 3.11 describes the positions of the three Quartiles. Figure 3.11 Quartiles. Q 2 Q 1 Q 3 Each of these portions contains 25% of the observations of a data set arranged in increasing order 25% 25% 25% 25% Approximately 25% of the values in a ranked data set are less than Q 1 and about 75% are greater than Q 1 . The second quartile, Q 2 , divides a ranked data set into two equal parts; hence, the second quartile and the median are the same. Approximately 75% of the data values are less than Q 3 and about 25% are greater than Q 3 . The difference between the third quartile and the first quartile for a data set is called the interquartile range (IQR), which is a measure of dispersion. Calculating Interquartile Range The difference between the third and the first Quartiles gives the interquartile range; that is, IQR = Interquartile range = Q 3 − Q 1 Examples 3–23 and 3–24 show the calculation of the Quartiles and the interquartile range. EXAMPLE 3–23 Commuting Times for College Students A sample of 12 commuter students was selected from a college. The following data give the typical one-way commuting times (in minutes) from home to college for these 12 students. 29 14 39 17 7 47 63 37 42 18 24 55 (a) Find the values of the three Quartiles.
eBook - PDF- Prem S. Mann(Author)
- 2017(Publication Date)
- Wiley(Publisher)
Figure 3.11 describes the positions of the three Quartiles. Figure 3.11 Quartiles. Q 2 Q 1 Q 3 Each of these portions contains 25% of the observations of a data set arranged in increasing order 25% 25% 25% 25% Approximately 25% of the values in a ranked data set are less than Q 1 and about 75% are greater than Q 1 . The second quartile, Q 2 , divides a ranked data set into two equal parts; hence, the second quartile and the median are the same. Approximately 75% of the data values are less than Q 3 and about 25% are greater than Q 3 . The difference between the third quartile and the first quartile for a data set is called the interquartile range (IQR), which is a measure of dispersion. Calculating Interquartile Range The difference between the third and the first Quartiles gives the interquartile range; that is, IQR = Interquartile range = Q 3 − Q 1 Examples 3–23 and 3–24 show the calculation of the Quartiles and the interquartile range. EXAMPLE 3–23 Commuting Times for College Students Finding Quartiles and the interquartile range. A sample of 12 commuter students was selected from a college. The following data give the typical one-way commuting times (in minutes) from home to college for these 12 students. 29 14 39 17 7 47 63 37 42 18 24 55 (a) Find the values of the three Quartiles. (b) Where does the commuting time of 47 fall in relation to the three Quartiles? (c) Find the interquartile range. Quartiles are three values that divide a ranked data set into four equal parts. The second quartile is the same as the median of a data set. The first quartile is the median of the observations that are less than the median, and the third quartile is the median of the observations that are greater than the median. 3.5 Measures of Position a. Using the empirical rule, find the (approximate) percentage of all college textbooks with their prices between i.
- Aliakbar Montazer Haghighi, Indika Wickramasinghe(Authors)
- 2020(Publication Date)
- CRC Press(Publisher)
C C 1 2 , , , C 99 are called the first , second , … , ninety-ninth per-centile , respectively. As a special case of percentile, we define the decile. If the sorted sample data, S , is divided into 10 equal-sized parts, each of the 9 parts, say D i i , 1 = ,2, ,9, is called a decile . Thus, each decile contains 10% of the data points (observations) contained in S . 329 Descriptive Statistics ⎛ n + 1 ⎞ In other words, D 1 = the value of 1 ⎜ ⎟ th data point, D 2 = the value of ⎝ 10 ⎠ ⎡ ⎛ n + 1 ⎞ ⎤ ⎡ ⎛ n + 1 ⎞ ⎤ ⎢ 2 ⎜ ⎟ ⎥ th data point, D 3 = the value of ⎢ 3 ⎜ ⎟ ⎥ th data point, …, D ⎣ ⎝ 10 ⎠ ⎦ ⎣ ⎝ 10 ⎠ ⎦ 9 = the ⎡ ⎛ n + 1 ⎞ ⎤ value of ⎢ 9 ⎜ ⎟ ⎥ th data point. ⎣ ⎝ 10 ⎠ ⎦ 4.6.2 Q UARTILE Yet another special case of a percentile is the quartile . If the sorted sample data, S , is divided into 4 equal-sized parts, each of the 3 parts is called a quartile . Thus, the first quartile , denoted by Q 1 or the lower quartile , is the number that 25% of the lowest data points fall below. The second quartile (also the 50th percentile ) of the sample, denoted by Q 2 , is also called the median . This is the number that 50% of the lowest data points fall below. Similarly, the third quartile , denoted by Q 3 , also referred to as the higher quartile , is the number that 75% of the lowest data points fall below and 25% fall above. In a set of data points , S , the interquartile range , denoted by IQR , is a mea-sure of where the “middle fifty” is. In other words, IQR is the range of the middle 50% of the data points. It is calculated as the difference between Q 3 and Q 1 , that is, Q 3 − Q 1 . As seen above, Quartiles are a special case of percentiles. We will illustrate how to calculate the percentiles and Quartiles in the following examples.
eBook - PDFMathematics NQF3 SB
TVET FIRST
- M Van Rensburg, I Mapaling, M Trollope A Thorne(Authors)
- 2017(Publication Date)
- Macmillan(Publisher)
Example 11.2 Two Mathematics classes achieved the following percentages in a test: Class 1: 15; 12; 20; 25; 99; 90; 89 Class 2: 49; 50; 48; 57; 44; 50; 52 What is the mean mark received in each class? Did each class get similar marks or was there a lot of variation? Solution: If you calculate the mean for each class, you will see that both classes have a mean of 50. However, if you look at the individual marks for each class, you will see that the first class had four learners who performed very poorly and three who performed very well. In comparison, the learners in the second class all achieved marks that were quite close to 50%. Measures of central tendency are not always enough to describe the data well; this is why we also look at measures of dispersion when describing data. Examples of measures of dispersion are the range, the Quartiles and the inter-quartile range, as shown in Table 11.2. Table 11.2: Measures of dispersion Measure Explanation and formula Range The difference between the biggest and smallest values R = x max − x min Quartiles Values that divide a set of data into equal parts. A data set has three Quartiles. Lower quartile (Q 1 ) This is the median of the lower half of the data. Its position is indicated by the formula: Q 1 ’s position = P Q 1 = 1 __ 4 ( n + 1). The value of Q 1 will be greater than a quarter of the values in the set (25%). Median (Q 2 ) The median divides a data set into two equal parts. It can be indicated by Q 2 , and we can calculate its position using the formula: Q 2 ’s position = P Q 2 = 1 __ 2 ( n + 1). The value of Q 2 will be greater than half of the values in the set (50%). 293 Module 11 Measure Explanation and formula Upper quartile (Q 3 ) This is the median of the upper half of the data. Its position is indicated by the formula: Q 3 ’s position = P Q 3 = 3 __ 4 ( n + 1). The value of Q 3 will be greater than 3 __ 4 of all of the values in the set (75%).
eBook - PDFStatistics
Learning from Data
- Roxy Peck, Tom Short(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
The ordered data values are 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 2 3 5 11 The median is the average of the two middle values. Here, both middle values are 0, and so the median is also 0. Dividing the data set into the lower and upper halves and then finding the median of each half leads to the following Quartiles. Lower half 0 0 0 0 0 0 0 0 0 0 0 0 lower quartile 5 0 Upper half 0 0 0 0 0 1 1 1 2 3 5 11 upper quartile 5 1 You can now calculate the value of the interquartile range. iqr 5 upper quartile 2 lower quartile 5 1 2 0 5 1 Notice here that the lower quartile and the median are both 0. This is because more than half of the observations in the data set have a value of 0. For this data set, the smallest 25% are all 0, and even the next 25% in the ordered list are all 0. One way to see this is to take another look at the ordered data list: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 2 3 5 11 25% of observations 25% of observations 25% of observations 25% of observations middle 50% The small value of the interquartile range indicates that there is relatively little variability in the middle half of the data distribution. Example 3.10 Some Strange Things Can Happen with Quartiles… Data set available Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. CHAPTER 3 Numerical Methods for Describing Data Distributions 138 These are the same steps that were used when describing center and variability of a data dis-tribution that is approximately symmetric.
eBook - PDF- Ned Freed, Stacey Jones, Timothy Bergquist(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
The theme throughout remains the same: transforming data into useful information. 3.1 Percentiles and Quartiles In Chapter 2 we saw how location indicators like the mean and the median can be used to summarize data. There are other location indicators that can also provide a quick snapshot of data set characteristics. We’ll consider two of them here. Percentiles According to recent statistics, 80% of the flights out of LAX International Airport are no more than 12 minutes late taking off, 50% of the students who take the SAT exam score 1520 or less, and 33% of the season ticket holders at the Metropolitan Opera have an annual income of no more than $38,000. All of these numbers—12, 1520, and 38,000—have one thing in common: they’re all percentiles. In data sets that don’t contain a large number of repeated values, the relative position of any value in the data set can be usefully defined in terms of its percentile . To illustrate, if 15% of the values in a given data set are 106 or less, 106 is said to be the 15 th percentile for that data set. If 77% of the values in the data set are 189 or less, 189 is said to be the 77 th percentile. In our examples above, then, 12 minutes can be described as the 80 th percentile for LAX late departure times, 1520 is the 50 th percentile for SAT scores, and $38,000 is the 33 rd percentile for season ticket-holder incomes. In general, for any given data set, the p th percentile is the value that bounds (approxi- mately) the lower p% of all the values in the data set. Put a little more formally, Percentiles If the value A is the pth percentile value for a data set, then at least p% of the values are less than or equal to A and at least (1-p)% of the values are greater than or equal to A. Although the idea of a percentile seems easy enough, finding percentile values can be a little tricky. The first step is to arrange the values in the data set in ascending order.
- Teresa Bradley(Author)
- 2014(Publication Date)
- Wiley(Publisher)
(b) Estimate each quartile approximately Solution (a) The calculations based on the data in Table SK1 5.2 are shown in Table 2.5. The lower quartile is the value of the item that is positioned 25 % of the way through the ordered data, i.e. at or nearest to 0.25(49 + 1)th = 12.5th item in the ordered data. From the cumulative frequency column in Table 2.5 the 12.5th item is in the interval (20 < 25). Since there are seven items in the first interval a further 5.5 items are required to reach the lower quartile value: this would be just less than half of the 12 items in this group, hence the value of Q 1 is 22.5 approximately See Table 2.5. The middle quartile or median is the value positioned half way through the ordered data, i.e. at or nearest the item in the 0.5(49 + 1)th = 25th position. The 25th item is in interval (30 < 35). Since there is only one item in this group, use the mid-interval value as an estimate of the median, so 32.5, as indicated in Table 2.5. The upper quartile is the value positioned 75 % through the ordered data, that is at or nearest to the 0.75(49 + 1)th = 37.5th item. This item is in the interval (40 < 45). To reach the 37.5th item requires the first 3.5 out of the 10 items in this group – approximately 1/3 of the way into the interval (40 < 45). Hence the value of the upper quartile ( Q 3 ) is 42 approximately (see Table 2.5). Table 2.5 Identifying the intervals containing the median and Quartiles Intervals (hours) Frequency: f i Less than Cumulative frequency 15 < 20 7 < 20 7 20 < 25 12 < 25 19 25 < 30 5 < 30 24 30 < 35 1 < 35 25 35 < 40 9 < 40 34 40 < 45 10 < 45 44 45 < 50 1 < 50 45 50 < 55 1 < 55 46 55 < 60 3 < 60 49 Q 1 the 12.5th item is in this interval: The value of Q 1 is 22.5 approximately Q 2 (median), the 25th item is in this interval: The value of Q 2 is 32.5 approximately Q 3 the 37.5th item is in this interval; the value of Q 3 is 42 approximately Compare the results for the raw data calculated in Worked Example 2.1.
- William DeCoursey(Author)
- 2003(Publication Date)
- Newnes(Publisher)
An order was obtained for fine powder. Although the specifications for this fine powder were within the size range of powder which had been produced in the past, the engineers in the plant found that very little of the powder produced at their best Descriptive Statistics: Summary Numbers 51 guess of the optimum conditions would satisfy the specifications. Thus, the mean size of the specification was satisfactory, but the specified variability was not satisfactory from the point of view of production. To make production of fine powder more practical, it was necessary to change the specifications for “fine powder” to correspond to a larger standard deviation. When this was done, the plant could produce fine powder much more easily (but the customer was not willing to pay such a large premium for it!). 3.3 Quartiles, Deciles, Percentiles, and Quantiles Quartiles, deciles, and percentiles divide a frequency distribution into a number of parts containing equal frequencies. The items are first put into order of increasing magnitude. Quartiles divide the range of values into four parts, each containing one quarter of the values. Again, if an item comes exactly on a dividing line, half of it is counted in the group above and half is counted below. Similarly, deciles divide into ten parts, each containing one tenth of the total frequency, and percentiles divide into a hundred parts, each containing one hundredth of the total frequency. If we think again about the median, it is the second or middle quartile, the fifth decile, and the fiftieth percentile. If a quartile, decile, or percentile falls between two items in order of size, for our purposes the value halfway between the two items will be used. Other conventions are also common, but the effect of different choices is usually not important. Remember that we are dealing with a quantity which varies randomly, so another sample would likely show a different quartile or decile or percentile.
- David Anderson, Dennis Sweeney, Thomas Williams, Jeffrey Camm(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
Copyright 2020 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 3.1 Measures of Location 115 The third quartile, or 75th percentile, is .75 of the way between the value in position 9 (5950) and the value in position 10 (6050). Thus, Q 3 5 5950 1 .75(6050 2 5950) 5 5950 1 .75(100) 5 6025 We defined the Quartiles as the 25th, 50th, and 75th percentiles. Thus, we computed the Quartiles in the same way as percentiles. However, other conventions are sometimes used to compute Quartiles, and the actual values reported for Quartiles may vary slightly depending on the convention used. Nevertheless, the objective of all procedures for computing quart-iles is to divide the data into four equal parts. Using Excel to Compute Percentiles and Quartiles Excel provides functions for computing percentiles and Quartiles. We will illustrate the use of these functions by showing how to compute the p th percentile and the Quartiles for the starting salary data in Table 3.1. Refer to Figure 3.4 as we describe the tasks involved. The formula worksheet is in the background; the value worksheet is in the foreground. Enter/Access Data: Open the file StartingSalaries . The data are in cells B2:B13 and labels are in column A and cell B1. Enter Functions and Formulas: Excel’s PERCENTILE.EXC function can be used to compute the p th percentile.
- Andrew F. Hayes(Author)
- 2020(Publication Date)
- Routledge(Publisher)
A box plot contains information about the median of a distribution, the interquartile range (IQR), the measurement interval that contains the inner 50% of measurements, and the minimum and maximum mea-surements in a distribution, while at the same time highlighting measurements that extremes TV Viewing Hours 12 11 10 9 8 7 6 5 4 3 2 1 0 Median Median + 1.5 IQR (or maximum, whichever is smallest) Median - 1.5 IQR Inner 50% outliers (or minimum, whichever -1 is largest) -2 61 4.6. Standardization Figure 4.4 A box plot of the TV viewing data. are unusual using certain criteria. A box plot of the TV viewing data is displayed in Figure 4.4. The figure itself is fairly self-explanatory. The dark line dividing the gray box is the median, while the upper and lower edges of the box define the end points of the ordinal middle 50% of the measurements. From the box plot, you can see that the median measurement is 2, whereas 50% of the measurements reside between 1.5 and 3. By definition, then, the interquartile range is 3 − 1 . 5 = 1 . 5. The long horizontal lines above and below the box are set at the median plus and minus 1.5 interquartile ranges. However, if the median plus 1.5 IQRs exceeds the maximum measurement, then the upper line is placed at the maximum. If the median minus 1.5 IQRs is smaller than the minimum measurement, then the lower line is set at the minimum. The box plot also depicts the “unusual” measurements, defined as those with measurements that are more than 1.5 IQRs from the median (in either direction). Different statistical programs will depict unusual cases differently. In SPSS (which generated this figure), “outliers” in a box plot are defined as cases with measurements between 1.5 and 3 IQRs from the median. “Extreme values” are defined by SPSS as measurements more than three IQRs from the median.
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