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.
6 Key excerpts on "Cumulative Frequency"
eBook - ePub- Debbie L. Hahs-Vaughn, Richard Lomax(Authors)
- 2020(Publication Date)
- Routledge(Publisher)
...At the same time, an ungrouped frequency distribution for that data would not have much of a message for the reader. Ultimately, the decisive factor is the adequacy with which information is communicated to the reader. There are no absolute rules on how to best group values into intervals. The nature of the interval grouping comes down to whatever form best represents the data. With today’s powerful statistical computer software, it is easy for the researcher to try several different interval widths before deciding which one works best for a particular set of data. Note also that the frequency distribution can be used with variables of any measurement scale, from nominal (e.g., the frequencies for eye color of a group of children) to ratio (e.g., the frequencies for the height of a group of adults). 2.1.2 Cumulative Frequency Distributions A second type of frequency distribution is known as the Cumulative Frequency distribution (cf). For the example data, this is depicted in the third column of Table 2.2 and labeled as “cf.” To put it simply, the number of cumulative frequencies for a particular interval is the number of scores contained in that interval and all of the smaller intervals. Thus, the 9 interval contains one frequency and there are no frequencies smaller than that interval, so the Cumulative Frequency is simply 1. The 10 interval contains one frequency and there is one frequency in a smaller interval, so the Cumulative Frequency is 2 (i.e., 1 + 1). The 11 interval contains two frequencies and there are two frequencies in smaller intervals; thus the Cumulative Frequency is 4 (i.e., 2 + 2). Then, four people had scores in the 11 interval and smaller intervals. One way to think about determining the Cumulative Frequency column is to take the frequency column and accumulate downward (i.e., from the top down, yielding 1; 1 + 1 = 2; 1 + 1 + 2 = 4; etc.)...
eBook - ePubIntroductory Probability and Statistics
Applications for Forestry and Natural Sciences (Revised Edition)
- Robert Kozak, Antal Kozak, Christina Staudhammer, Susan Watts(Authors)
- 2019(Publication Date)
- CAB International(Publisher)
...The Cumulative Frequency is the frequency of all observations less than the upper class boundary of a given class. It answers the question, ‘How many observations fall within a certain class or lower?’, and is often referred to as the ‘less than frequency’. Conversely, the inverse Cumulative Frequency represents the frequency of all values greater than the lower class boundary of a given class. It answers the question, ‘How many observations fall within a certain class or higher?’, and is often referred to as the ‘more than frequency’. Table 2.7 shows how the cumulative and the inverse cumulative frequencies are calculated. For the Cumulative Frequency, the frequency of each class is added to the number of observations that fall below that class, starting with the first class. The same logic is applied to the inverse Cumulative Frequency, but starting with the last class. Both cumulative frequencies can be expressed as percentages (or proportions) of the total frequencies and, in these cases, are called relative cumulative frequencies and inverse relative cumulative frequencies, respectively. We may also be interested in categorizing data from Table 1.1 (see Chapter 1) in terms of the pattern or shape of two variables. Table 2.8 shows a bivariate frequency distribution, which gives the distribution of trees by species and number of neighbouring trees. In constructing Table 2.8, we use a categorical frequency distribution in one direction and an ungrouped frequency distribution in the other. Any combination of the three frequency distributions (categorical, ungrouped and grouped) can be combined to form bivariate frequency distributions. Table 2.6. Preparation of grouped frequency distribution of 50 dbh measurements. Table 2.7. Preparation of cumulative and inverse Cumulative Frequency distributions of 50 dbh measurements. Table 2.8. Bivariate frequency distribution of 50 trees...
eBook - ePubStatistics for the Behavioural Sciences
An Introduction to Frequentist and Bayesian Approaches
- Riccardo Russo(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...The percentage Cumulative Frequency of the scores is also useful since it can be used to obtain the percentage of observations falling at or below a given score. Table 2.1 Hypothetical raw data on the number of pictures named in a sample of 40 4-year-old children. Their frequency distribution is also provided From the frequency distribution table, it can be easily seen what are the lowest and the highest naming scores (i.e., 12 and 42, respectively), and what is the most common score (i.e., 26). Using the Cumulative Frequency column, it can be seen that 10 children could name 21 pictures or less. From the percentage Cumulative Frequency column, it can be observed that a score of up to and including 21 was obtained by 25% of the children assessed. Frequency distributions can be plotted in a pictorial format called a histogram or bar chart. In the naming data case, the number of the named pictures is marked along the horizontal axis. The height of the vertical bars indicates the frequency of occurrence of each naming score (see Figure 2.1). Figure 2.2 displays the Cumulative Frequency distribution of the data presented in Table 2.1. I noticed that students often find difficult to grasp what Cumulative Frequency distributions depict. If you do share the same, then I suggest you to check for data usage in the setting of your smartphone. You will see that a graph is depicted indicating the progressive amount of data used by your phone in a limited period of time (e.g., 1 week). This graph is somehow comparable to a Cumulative Frequency distribution. Figure 2.1 Histogram displaying the data from Table 2.1. Figure 2.2 Cumulative Frequency histogram of the data in Table 2.1. 2.3 Grouped data In the naming data, there were some scores with a frequency of 0 and several other scores with frequencies no higher than 2...
eBook - ePub- Perumal Mariappan(Author)
- 2019(Publication Date)
- Chapman and Hall/CRC(Publisher)
...It shows the percentage for each class. It helps to understand the concept of probability and to compare 2 or more sets of data. Example : Form the frequency distribution for the following data given weights in pounds of 30 college students: Take class intervals of 10 units each. Also, construct the Cumulative Frequency and relative frequency distribution. Given the length of the class interval = 10 Minimum weight = 104 ; Maximum weight = 168 Range = 166 − 104 = 62 Number of intervals = range / length = 62 / 10 = 6.2 = 7 approximately Weight (lbs.) Tally Marks Frequency Cumulative Frequency Relative Frequency 100–110 \\ 2 2 2/30 = 0.067 110–120 \\\ 3 2 + 3 = 5 3/30 =. 0.100 120–130 \\\\ 5 5 + 5 = 10 5/30 = 0.167 130–140 \\\\ \\\\ 10 10 + 10 = 20 10/30 = 0.333 140–150 \\\\ \ 6 20 + 6 = 26 6/30 = 0.200 150–160 \\\ 3 26 + 3 = 29 3/30 = 0.100 160–170 \ 1 29 + 1 = 30 1/30 = 0.033 Total 30 1.000 The Cumulative Frequency column helps find how many elements are up to that class interval. In the preceding example, there are 5 elements up to 120 pounds. The relative frequency column helps to find the percentage of elements present in that interval. In the example, 20% items fall in the interval (150–160) pounds. 3.12 Diagrammatic Representation of Data Usually the statistical data can be presented in the form of statements and tables; additionally, it can also be represented in the form of diagrams. Diagrams are ideal visual methods of presenting data. It helps people understand the data easily. Engineers, managers, technical men, and businessmen have been using them for a long time. What is hidden in a mass of data is brought out clearly and within a second, we get a cross-sectional diagram of the whole situation. A diagram of daily or weekly sales tells the manager quickly the trend of the business. Government sectors also use the diagram to show the nation’s economic development...
eBook - ePub- Mark Patmore(Author)
- 2018(Publication Date)
- Learning Matters(Publisher)
...4 Interpreting and using written data Notes You will not be required to draw tables or plot graphs when answering written data questions. The following notes are intended to give you a brief summary of some aspects, including terms and representations that may be unfamiliar. Some of the information received by schools, for example, analyses of pupil performance, uses ‘cumulative frequencies’ or ‘cumulative percentages’. One way to illustrate cumulative frequencies is through an example. The table shows the marks gained in a test by the 60 pupils in a year group. We could complete a tally chart and a frequency table. But 60 results are a lot to analyse and we could group the results together in intervals. A sensible interval in this case would be a band of 10 marks. This is a bit like putting the results into ‘bins’. Note that ≤ means ‘less than or equal to’ and < means ‘less than’, so 30 ≤ m < 40 means all the marks between 30 and 40 including 30 but excluding 40. Here are the marks grouped into a frequency table. The last column ‘Cumulative Frequency’ gives the ‘running total’ – in this case the number of pupils with less than a certain mark. For example, there are 38 pupils who gained less than 40 marks. The values for Cumulative Frequency can be plotted to give a Cumulative Frequency curve as shown below. Note that the Cumulative Frequency values are plotted at the right-hand end of each interval, i.e. at 10, 20, 30 and so on. You can use a Cumulative Frequency curve to estimate the median mark: the median for any particular assessment is the score or level which half the relevant pupils exceed and half do not achieve. There are 60 pupils, so the median mark will be the 30th mark. (Find 30 on the vertical scale and go across the graph until you reach the curve and read off the value on the horizontal scale.) The median mark is about 37 – check that you agree. It is also possible to find the quartiles...
eBook - ePub- Joseph Patton(Author)
- 2004(Publication Date)
- International Society of Automation(Publisher)
...The percentages are determined by dividing each number in Table 11-3 by the total of 119. For example, 9/119 = 0.0756, which rounds to 8%. Table 11-4 Cumulative percentages show progress toward 100 percent. 8 10 20 33 40 46 51 66 74 81 85 90 100 The cumulative distribution on ordered data is very useful for determining the probability of need for each amount of time. This is very useful in scheduling PM, planning parts, and for other maintenance predictions. Table 11-5 shows the cumulative percentages for the Table 11-2 ordered data. It can be seen from the data that filling 85% of the requests requires 15 minutes. Table 11-5 Relationships show when cumulative percentages are ordered by time. MEAN, MODE, AND MEDIAN Mean is the statistical term for arithmetic average. In our specific case, that would add all the time required for the work orders (119 minutes), and divide them by the number of weeks, which is 13 in our example. The result is 119 13 = 9.15, which is about 9. The mean tells us the average of all the usage. That number conveys a lot of information. Be alert, though, to the fact that a person can stand with one foot in a 120-degree oven and the other foot in a 20-degree freezer and on the average would be at a 70-degree temperature, but the person would not be comfortable. Mode is the most frequently occurring number, in this case “9.” In the PM business, a frequently occurring mode probably indicates that work is done in a consistent pattern. The median is the midpoint of the distribution. In our example of 13 numbers, look at the ordered data and count to the seventh number; that is the median, which is also 9. GRAPHICS People gain information more easily from graphical displays than from lists of pure numbers. PM analysts will find frequent use for graphs and plots. Most analysts simply enter the data into a computer program and it will display the graphical plot on the monitor screen or printout...
