Mathematics
Cumulative Frequency
Cumulative frequency refers to the running total of frequencies in a data set. It is used to show the total number of observations that lie below a certain value in a frequency distribution. This concept is often represented graphically using a cumulative frequency curve or ogive, which helps in analyzing the distribution of data.
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11 Key excerpts on "Cumulative Frequency"
eBook - PDF- Dawn A. Willoughby(Author)
- 2016(Publication Date)
- Wiley(Publisher)
• Cumulative Frequency represents the number of data values we have collected that belong to that class or the classes that come before it in the table; it is a running total of the class frequencies. • Cumulative relative frequencies allow us to express cumulative frequencies as a cumulative percentage of the total number of data values. Although a grouped frequency distribution is a useful way of summarising a data set, it does have one disadvantage: the individual values from the data set can no longer be identified by looking at the table. The grouped frequency distribution only shows the number of data values that belong to each class. We can still determine characteristics of the data set and highlight 50 A N E S S E N T I A L G U I D E T O B U S I N E S S S T A T I S T I C S important features or underlying patterns, but we are not able to comment on individual pieces of data that have been collected. EXAMPLE – DISCRETE DATA Suppose we recorded the number of boarding gates for passengers at 75 international airports. Our raw data might be presented as follows: 114 46 43 61 38 21 75 40 94 51 96 10 92 19 17 39 70 27 46 114 35 68 81 114 74 95 38 17 14 74 20 108 76 28 108 45 56 38 36 17 46 17 103 109 41 88 125 14 121 44 106 55 44 38 17 16 40 85 34 103 101 66 64 50 86 61 60 42 99 59 19 33 50 38 110 The smallest data value is 10 and the largest is 125, so a frequency distribution would give the table below. It can be seen that this is not a suitable way to summarise our information. Number of passenger boarding gates Tally Frequency 10 j 1 11 12 13 14 j j 2 15 . . . . . . . . . . . . 120 121 j 1 122 123 3 D A T A D I S T R I B U T I O N S 51 (Continued ) Number of passenger boarding gates Tally Frequency 124 125 | 1 Total 75 Choosing appropriate classes, the grouped frequency distribution is shown below; this is a more appropriate distribution for our data.
eBook - PDF- Lorie Darche, Gerda Koch(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
Using this table, it can be noted that the number of requests totaled 10 or more in that month 90% of the time. EXAMPLE: Listed are the final test scores for 100 students at Studi-ous State University (Table 2.6). The recorded scores are as follows: 58 68 73 61 66 96 79 65 86 69 94 84 79 80 65 78 78 62 80 67 74 97 75 88 75 82 77 89 67 73 73 83 82 82 73 87 75 61 97 74 57 81 81 69 68 60 74 94 75 78 88 72 75 88 85 90 93 62 77 95 85 78 63 94 92 71 62 71 95 69 60 76 62 76 84 92 88 59 60 78 74 79 65 76 75 92 84 76 85 63 68 72 83 71 53 85 96 95 93 75 Highest score 5 97; lowest score 5 53; range 44 (97 53 44) 5 2 5 Ranked Listing: 97 97 96 96 95 95 95 94 94 94 93 93 92 92 92 90 89 89 88 88 88 88 87 86 85 85 85 85 84 84 84 83 83 82 82 82 81 81 80 80 79 79 78 78 78 78 78 77 77 76 76 76 76 75 75 75 75 75 75 75 74 74 74 74 73 73 73 73 72 72 71 71 71 69 69 69 68 68 68 67 67 66 65 65 65 63 63 62 62 62 62 61 61 60 60 60 59 58 57 53 h. Cumulative Frequency The Cumulative Frequency is the sum of the frequencies, starting at the lowest interval and including the frequencies within that inter-val. This column is prepared by adding in successive class frequencies from the bottom to the top. The entry opposite the lowest interval is the frequency in that interval; the entry opposite the second interval is the sum of the frequencies in the first and second intervals; the entry opposite the third interval is the sum of the frequencies in the first, second, and third intervals; and so on. The entry opposite the top interval equals the total number (N) in the distribution. Cumulating frequencies is most commonly done from bottom to top, but it is also possible to cumulate from the top downward. 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).
eBook - ePub- Frederick Mosteller, Stephen E. Fienberg, Robert E.K. Rourke, Stephen E. Fienberg, Robert E.K. Rourke(Authors)
- 2013(Publication Date)
- Dover Publications(Publisher)
1. How many measurements are there at a given value or in a given set of values (interval)? We get the answer directly from the table or histogram. This tells us where the measurements are concentrated and where they are sparse.2. What proportion or percentage of the measurements is less than some given number? We can get the answer by adding group frequencies, but we get it more readily from cumulative frequencies when we have them. These tell us the proportions less than or greater than various numbers: What fraction of students finish at least the 11th grade of secondary school?To get cumulative frequencies, we add, or accumulate, the group frequencies as we go from the group of smallest measurements to the group of largest measurements. Thus the frequencies of lengths of pipes required for building materials shown in Table 1-4a yield the cumulative frequencies in Table 1-4b .TABLE 1-4Frequency distributions for a set of40 measurements (in feet) of lengths of pipesTABLE 1-4(a)Measurement interval Frequency 3.0-3.9 13 4.0-4.9 23 5.0-5.9 3 6.0-6.9 1 Total 40 TABLE 1-4(b)Measurements less than Cumulative Frequency Percent† 3.0 0 0 4.0 13 32.5 5.0 36 90 6.0 39 97.5 7.0 40 100 † Note: We get the percent by dividing the Cumulative Frequency by the total, 40, and multiplying by 100.TABLE 1-4(c)Measurements greater than or equal to Cumulative Frequency Percent 7.0 0 0 6.0 1 2.5 5.0 4 10 4.0 27 67.5 3.0 40 100 Cumulative Frequencies and Their Reverses. In Table 1-4b we began the cumulating with the group of smallest measurements and cumulated toward the group of largest measurements. We can get another set of cumulative frequencies by working in the opposite direction, that is, by beginning with the group of largest measurements and cumulating toward the group of smallest measurements. This new Cumulative Frequency distribution (Table 1-4c ) is the complement of the one in Table 1-4b , which is cumulated from the smallest measurements to the largest. For example, the Cumulative Frequency in Table 1-4b for measurements less than 4 is 13, and in Table 1-4c the Cumulative Frequency for measurements greater than or equal to 4 is 40 – 13 = 27. The corresponding cumulative percentages are 32.5 and 100 – 32.5 = 67.5. We can call either Table 1-4b or Table 1-4c the Cumulative Frequency distribution; then the other one is the reverse cumulative distribution. Hence, if Table 1-4b is called the Cumulative Frequency distribution, then Table 1-4c
eBook - PDFBiostatistics
Basic Concepts and Methodology for the Health Sciences, 10th Edition International Student Version
- Wayne W. Daniel, Chad L. Cross(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
In determining the frequency of values falling within two or more class intervals, we obtain the sum of the number of values falling within the class intervals of interest. Similarly, if we want to know the relative frequency of occurrence of values falling within two or more class intervals, we add the respective relative frequencies. We may sum, or cumulate, the frequencies and relative frequencies to facilitate obtaining information regarding the frequency or relative frequency of values within two or more contiguous class intervals. Table 2.3.2 shows the data of Table 2.3.1 along with the cumulative frequencies, the relative frequencies, and cumulative relative frequencies. Suppose that we are interested in the relative frequency of values between 50 and 79. We use the cumulative relative frequency column of Table 2.3.2 and subtract .3016 from .9948, obtaining .6932. We may use a statistical package to obtain a table similar to that shown in Table 2.3.2. Tables obtained from both MINITAB and SPSS software are shown in Figure 2.3.1. The Histogram We may display a frequency distribution (or a relative frequency distribution) graphically in the form of a histogram, which is a special type of bar graph. When we construct a histogram the values of the variable under consideration are represented by the horizontal axis, while the vertical axis has as its scale the frequency (or relative frequency if desired) of occurrence. Above each class interval on the horizontal axis a rectangular bar, or cell, as it is sometimes called, is erected so that the height corresponds to the respective frequency when the class intervals are of equal width. The cells of a histogram must be joined and, to accomplish this, we must take into account the true boundaries of the class intervals to prevent gaps from occurring between the cells of our graph. The level of precision observed in reported data that are measured on a continuous scale indicates some order of rounding.
eBook - PDF- M Smit R Jonker(Author)
- 2017(Publication Date)
- Macmillan(Publisher)
2. When the frequencies of values in a data set are listed in a table, the table is known as a frequency distribution table and the list is referred to as a frequency distribution . 3. Discrete data is generated by counting. We count how many times a particular value is repeated and the number of times a number is repeated is known as its frequency. 4. A data class is a group of data that is related by some user defined property; e.g. age classes such as 8–9; 10–12; 13–14 years, and so on; each of these groups is called a class . 5. A class interval is the numerical width or the range of values in each class; also referred to as class size . 132 Module 12 Summary continued 6. Cumulative Frequency distribution is the sum of a particular class and all the classes below it in a frequency distribution. To calculate Cumulative Frequency we add each frequency to the sum of the preceding frequencies. The last value will always be equal to the total for all observations. 7. A bar chart or bar graph presents grouped data by means of rectangular bars whose lengths are proportional to the values they represent. 8. Measures of central tendency are numbers that describe what is average or typical within a distribution of data. The three main measures of central tendency are mode, median and mean. 9. Always sort the data from smallest to biggest before calculating measures of central tendency or variability. 10. The mode of a data collection is the element that tends to occur most often in the frequency table. 11. Median is the middle value of a data set. To find the median, we use the formula: 1 Median 2 n + = where n is the number of data items in the data range. 12. If the data set consists of an even number of data values the median formula will give us the position of the median. The median will be the mean (average) of the two middle values.
- Joseph Healey, Christopher Donoghue, Joseph Healey(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
Cumulative Frequency Column 1. Begin with the lowest class interval (the interval with the lowest scores). The entry in the Cumulative Frequency column will be the same as the number of cases in this interval. In this case, the value for both columns is 11. 2. Go to the next class interval. The Cumulative Frequency for this interval is the number of cases in the interval plus the number of cases in the lower interval, or 11 1 5 5 16. 3. Continue adding (or accumulating) cases from interval to interval until you reach the interval with the highest scores, which will have a Cumulative Frequency equal to N. Cumulative Percentage Column 1. Compute the percentage of cases in each category and then follow the pattern for the cumulative frequencies. The entry for the lowest class interval will be the same as the percentage of cases in the interval. For this table, both the percentage and cumulative percentage for the first interval are equal to 55%. 2. For the next higher interval, the cumulative percentage is the percentage of cases in the interval plus the percentage of cases in the lower interval. So, 55.0% 1 25% 5 80.0%. 3. Continue adding (or accumulating) percentages from interval to interval until you reach the interval with the highest scores, which will have a cumulative percentage of 100%. ONE STEP AT A TIME Adding Cumulative Frequency and Percentage Columns to Frequency Distributions Copyright 2021 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. 38 PART I DESCRIPTIVE STATISTICS are younger than 21.
- Laurent Seuront(Author)
- 2009(Publication Date)
- CRC Press(Publisher)
147 5 Frequency Distribution Dimensions A common procedure for looking at the level of organization of any data set is to study the prob-ability density function (PDF) or the cumulative density function (CDF). In particular, cumulative hypergeometric frequency distributions have been found in many areas of the natural sciences (see, for example, Laherrere and Sornette 1.998 for a review) and imply a wide range of values with many small values and few large values. This chapter focuses on several aspects of the cumulative fre-quency distribution of self-similar and self-affine patterns. This includes theoretical investigations of the correspondence between cumulative distribution functions and probability density functions (Section 5.1.) and the descriptions of the frequency distributions of intensities, areas, and volumes (Sections 5.2, 5.3., and 5.4). Special attention is finally given to rank-frequency distributions, from their original development in linguistics and their link to information theory and entropy to their effectiveness as a simple and direct diagnostic tool for ecologists to assess ecosystem complexity and their applicability to the analysis of symbolic sequences (Section 5.5). 5.1 CUMULATIVE DISTRIBUTION FUNCTIONS AND PROBABILITY DENSITY FUNCTIONS 5.1.1 T HEORY The Pareto law was originally introduced in economics to describe the number of people whose per-sonal incomes exceeded a given value (Pareto 1.896). More generally, the Pareto law of any random variable X is described in terms of the cumulative density function (CDF): P [ X ≥ x ] ∝ x − f (5.1.) where x is a threshold value, and f is the slope of a log-log plot of P [ X ≥ x ] vs. x . Note that Equation (5.1.) can be equivalently rewritten in terms of the PDF as (Faloutsos et al. 1.999): P [ X = x ] ∝ x − m (5.2) where m ( m = f + 1.) is the slope of a log-log plot of P [ X = x ] vs.
- Theodore Coladarci, Casey D. Cobb(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
To avoid the appearance of bias, arrange this list either alphabetically or by descending magnitude of frequency (as in Table 2.10). Step 2 Record the frequency, f, associated with each category and, if you wish, the corresponding percentage. Report the total number of cases, n, at the bottom of the frequencies column. Question: Would it be appropriate to include cumulative frequencies and percentages in this frequency distribution? Of course not, for it makes no sense to talk about a teacher “falling below” stickers or any other category of this qualitative variable (just as, in Chapter 1, it made no sense to claim that Asian is “less than” African American). Cumulative indices imply an underlying con- tinuum of scores and therefore are reserved for variables that are at least ordinal. 2.11 Summary Reading the Research: Quartiles As you saw in Section 2.9, quartiles refer to any of the three values (Q 1 , Q 2 , and Q 3 ) that separate a frequency distribution into four equal groups. In practice, however, the term quartile often is used to designate any one of the resulting four Table 2.10 Frequency Distribution for a Qualitative (Nominal) Variable Dominant Reward Strategy f % Verbal praise 21 70 Stickers 6 20 Privileges 3 10 n 30 It is difficult for data to tell their story until they have been organized in some fashion. Frequency distribu- tions make the meaning of data more easily grasped. Frequency distributions can show both the absolute frequency (how many?) and the relative frequency (what proportion or percentage?) associated with a score, class interval, or category. For quantitative vari- ables, the cumulative percentage frequency distribu- tion presents the percentage of cases that fall below a score or class interval. This kind of frequency distribu- tion also permits the identification of percentiles and percentile ranks. 26 Chapter 2 Frequency Distributions
- Robert Pagano(Author)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Percentiles 55 Table 3.7 shows the frequency distribution of statistics exam scores expressed as relative frequency, Cumulative Frequency, and cumulative percentage distributions. To convert a frequency distribution into a relative frequency distribution, the frequency for each interval is divided by the total number of scores. Thus, Relative f f N For example, the relative frequency for the interval 45–49 is found by dividing its frequency (1) by the total number of scores (70). Thus, the relative frequency for this interval 1 70 0.01. The relative frequency is useful because it tells us the proportion of scores contained in the interval. The Cumulative Frequency for each interval is found by adding the frequency of that interval to the frequencies of all the class intervals below it. Thus, the Cumulative Frequency for the interval 60–64 4 4 2 1 11. The cumulative percentage for each interval is found by converting cumulative frequencies to cumulative percentages. The equation for doing this is: cum% cum f N 100 For the interval 60–64, cum% cum f N 100 11 70 100 15.71 % Cumulative Frequency and cumulative percentage distributions are especially useful for finding percentiles and percentile ranks. PERCENTILES Percentiles are measures of relative standing. They are used extensively in education to compare the performance of an individual to that of a reference group. Thus, the Class Interval f Relative f Cumulative f Cumulative % 95–99 4 0.06 70 100 90–94 6 0.09 66 94.29 85–89 7 0.10 60 85.71 80–84 10 0.14 53 75.71 75–79 16 0.23 43 61.43 70–74 9 0.13 27 38.57 65–69 7 0.10 18 25.71 60–64 4 0.06 11 15.71 55–59 4 0.06 7 10.00 50–54 2 0.03 3 4.29 45–49 —– 1 0.01 1 1.43 70 1.00 table 3.7 Relative frequency, Cumulative Frequency, and cumulative percentage distributions for the grouped scores in Table 3.4 Copyright 2013 Cengage Learning.
eBook - PDF- Lorena Madrigal(Author)
- 2012(Publication Date)
- Cambridge University Press(Publisher)
Frequency Percent Cumulative Frequency Cumulative percent Group 1 50 50.0 50.0 50.0 2 50 50.0 100.0 100.0 Total 100 100.0 in the first group is 2 50 = 4% and in the second group is 7 50 = 14%. The percent column gives the reader important information: a family size of 12 is much more frequent in the second rather than the first group. The last two columns of a frequency distribution give you similar information, but in frequency and percent formats. How many women 2.1 Frequency distributions 15 produced five or fewer children in groups one and two? To answer this question we look at the Cumulative Frequency column: while 13 women produced five or fewer children in the first group, only four women did so in group two. In the same way, we can say that 26% of all women in the first group produced five or fewer children, while only 8% of all women did so in the second group. You can see that the last two columns provide very rich information about how the data are distributed in our two groups. The frequency distribution of both groups indicates that women in group one tend to produce fewer children than do women in group two. 2.1.2 Frequency distributions of continuous numeric variables Constructing a frequency distribution of discontinuous numerical and qualitative data is easy enough because it is easy to classify subjects since they have an identifiable observed variate: a woman has zero, one, two, or three living children. Even if a woman is not sure about how many surviving children she currently has, you can classify her into a category such as “indeterminate value.” In contrast, constructing a frequency dis- tribution of continuous numerical data is more complicated due to the precision involved in the measurement of the data. The problem of doing frequency distributions with con- tinuous data arises precisely during the process of assigning subjects to categories.
eBook - PDF- Samuel Stanley Wilks(Author)
- 2015(Publication Date)
- Princeton University Press(Publisher)
and cumulative grouped frequency distributions. To define these grouped distributions we first construct a frequency table. Returning to Table 2.1 (or looking at Figure 2.1) we find the least value in the table to be 1.32 and the largest value to be 1.77. The range is .45. We now divide the range into a number of equal intervals of convenient length. This means that the length should be a "round number". The number of intervals is usually taken to be between 10 and 25. A convenient interval for our example is 0.05, which gives us 10 class intervals or cells. We might also have used 0.04 , 0.03, or 0.02, but. we would usually avoid 0.0333, 0.035, and other such inconvenient numbers. We now take the cells to be 1.275 - 1.325, 1.325 - 1.375, 1.375 - 1.425, and so on to 1.725 - 1.775. Note the following features of these cells: (a) each is of length 0,05, (b) the boundaries of each cell end in a 5 and are written with one more decimal than is used in the original data of Table 2.1, (c) the upper boundary of any cell is the same as the lower boundary of the suc- ceeding cell (this is a convenience and will cause no ambiguity since the boun- daries are written to one more decimal than is used in the original data in Table 2.1). The cells are constructed so they will have the following simple mid- points respectively: 1.30, 1.35, 1.40, and so on to 1.75. (We can have all these nice properties since we took a round number for the cell length.) 20 2. FREQUENCY DISTRIBUTIONS Sec a 2.2 The frequency table can be exhibited as Table 2.2. The cell boundaries are shown in column (a), the midpoints in column (b). The tallied frequencies with which the observations fall into the various cells are shown in column (c). The frequencies are shown in column (d). The relative frequencies in column (e), the cumulative frequencies in column (f), and the cumulative relative frequencies in column (g).
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