cumulant

Examples of cumulant in the following topics:

• Cumulative Frequency Distributions

  • A cumulative frequency distribution displays a running total of all the preceding frequencies in a frequency distribution.
  • A cumulative frequency distribution is the sum of the class and all classes below it in a frequency distribution.
  • The third column should be labeled Cumulative Frequency.
  • There are a number of ways in which cumulative frequency distributions can be displayed graphically.
  • This image shows the difference between an ordinary histogram and a cumulative frequency histogram.

• Frequency Polygons

  • Frequency polygons are also a good choice for displaying cumulative frequency distributions.
  • A cumulative frequency polygon for the same test scores is shown in Figure 2.
  • Since 642 students took the test, the cumulative frequency for the last interval is 642.
  • It is also possible to plot two cumulative frequency distributions in the same graph.

• Relative Frequency Distributions

  • Just like we use cumulative frequency distributions when discussing simple frequency distributions, we often use cumulative frequency distributions when dealing with relative frequency as well.
  • Cumulative relative frequency (also called an ogive) is the accumulation of the previous relative frequencies.
  • To find the cumulative relative frequencies, add all the previous relative frequencies to the relative frequency for the current row.

• Frequency

  • Cumulative relative frequency is the accumulation of the previous relative frequencies.
  • To find the cumulative relative frequencies, add all the previous relative frequencies to the relative frequency for the current row.
  • The last entry of the cumulative relative frequency column is one, indicating that one hundred percent of the data has been accumulated.
  • This percentage is the cumulative relative frequency entry in the third row.
  • To find the cumulative relative frequency, add all of the previous relative frequencies to the relative frequency for the current row.

• Continuous Probability Distributions

  • Mathematicians also call such a distribution "absolutely continuous," since its cumulative distribution function is absolutely continuous with respect to the Lebesgue measure $\lambda$.
  • The definition states that a continuous probability distribution must possess a density; or equivalently, its cumulative distribution function be absolutely continuous.
  • This requirement is stronger than simple continuity of the cumulative distribution function, and there is a special class of distributions—singular distributions, which are neither continuous nor discrete nor a mixture of those.

• Optional Collaborative Classrom Exercise

  • In your class, have someone conduct a survey of the number of siblings (brothers and sisters) each student has.Create a frequency table.Add to it a relative frequency column and a cumulative relative frequency column.Answer the following questions:

• Summary

• Properties of Continuous Probability Distributions

  • Area under the curve is given by a different function called the cumulative distribution function (abbreviated: cdf).
  • The cumulative distribution function is used to evaluate probability as area.

• Practice 1: Goodness-of-Fit Test

  • The cumulative number of AIDS cases reported for Santa Clara County is broken down by ethnicity as follows: (Source: HIV/AIDS Epidemiology Santa Clara County, Santa Clara County Public Health Department, May 2011)

• Determining Sample Size

  • As follows, this can be estimated by pre-determined tables for certain values, by Mead's resource equation, or, more generally, by the cumulative distribution function.
  • Calculate the appropriate sample size required to yield a certain power for a hypothesis test by using predetermined tables, Mead's resource equation or the cumulative distribution function.