Frequency Distribution - Biostatistics - Ravinandan A P.pdf
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A frequency distribution arranges data into intervals and counts the number of observations in each interval. It displays grouped data with lower and upper class limits defining each interval. Relative and percentage frequency distributions show the ratio or percentage of observations in each interval rather than raw counts. Cumulative frequency distributions sum the frequencies of intervals up to each class limit. Frequency tables can summarize categorical data by interval counts, relative frequencies, and cumulative proportions. Cross tabulations examine frequencies of observations across multiple variables in a single table.
Frequency Distribution - Biostatistics - Ravinandan A P.pdf
- 1. Frequency Distribution Ravinandan A P Assistant Professor Sree Siddaganga College of Pharmacy In association with Siddaganga Hospital Tumkur-02
- 3. • A frequency distribution is a tabular arrangement of data whereby the data is grouped into different intervals, & then the number of observations that belong to each interval is determined. • Data that is presented in this manner are known as grouped data.
- 4. • The smallest value that can belong to a given interval is called the lower class limit • While the largest value that can belong to the interval is called the upper class limit. • The difference between the upper class limit & the lower class limit is defined to be the class width. • When designing the intervals to be used in a frequency distribution, it is preferable that the class widths of all intervals be the same.
- 5. • Relative frequency distribution & percentage frequency distribution are variants of the frequency distribution. • The relative frequency distribution is similar to the frequency distribution, except that instead of the number of observations belonging to a particular interval, the ratio of the number of observations in the interval to the total number of observations, also known as the relative frequency, is determined. • The % frequency distribution is arrived at by multiplying the relative frequencies of each interval by 100%.
- 6. Cumulative frequency distribution • It is obtained by computing the cumulative frequency, defined as the total frequency of all values less than the upper class limit of a particular interval, for all intervals. • From a frequency distribution, this can be done by simply adding together the frequencies of the interval and all other preceding intervals (i.e., intervals whose values are less than the values of a particular interval). • We can also calculate the relative cumulative frequency distribution and the percentage cumulative frequency distribution from the cumulative frequency distribution.
- 8. Frequency Table: Frequency or One-way Tables: • Frequency or one-way tables represent the simplest method for analyzing categorical data. • They are often used as one of the exploratory procedures to review how different categories of values are distributed in the sample. • For ex, in a survey of spectator interest in different physical exercise, we could summarize the respondents' interest in watching football in a frequency table as follows:
- 9. The table above shows the number, proportion, and cumulative proportion of respondents who characterized their interest in physical exercise as either (1) Always interested, (2) Usually interested, (3) Sometimes interested, or (4) Never interested. Category Count Cumulative Count Percent Cumulative Percent ALWAYS : Always interested USUALLY : Usually interested SOMETIMS: Sometimes interested NEVER : Never interested Missing 39 16 26 19 0 39 55 81 100 100 39.00000 16.00000 26.00000 19.00000 0.00000 39.0000 55.0000 81.0000 100.0000 100.0000
- 11. Two-way table: • This table shows two characteristics and is formed when either the caption or the stub is divided into two or more parts.
- 13. Cross tabulation: • Cross tabulation is a combination of two (or more) frequency tables arranged such that each cell in the resulting table represents a unique combination of specific values of cross tabulated variables. • Thus, cross tabulation allows us to examine frequencies of observations that belong to specific categories on more than one variable.
- 14. SODA: A SODA: B GENDER: MALE 20 (40%) 30 (60%) 50 (50%) GENDER: FEMALE 30 (60%) 20 (40%) 50 (50%) 50 (50%) 50 (50%) 100 (100%) The resulting cross tabulation could look as follows. Theoretically, an unlimited number of variables can be cross tabulated in a single multi-way table. However, research practice shows that it is usually difficult to examine and "understand" tables that involve more than 4 variables.
- 15. 2.5 5.9 3.2 1.4 7.0 4.3 8.9 0.7 4.2 9.9 3.4 4.6 5.0 6.4 1.1 9.2 7.7 0.9 4.0 2.3 5.6 2.2 3.1 4.7 5.5 6.6 1.9 3.9 6.1 5.2 8.2 3.3 2.2 5.8 4.1 3.8 1.2 6.8 9.5 0.8 Example 1: Given the following set of measurements for a particular sample: We note that the values range from 0 to 10.0. Therefore, we can create the following 10 classes: class 1: 0 - 1.0 class 6: 5.0 - 6.0 class 2: 1.0 - 2.0 class 7: 6.0 - 7.0 class 3: 2.0 - 3.0 class 8: 7.0 - 8.0 class 4: 3.0 - 4.0 class 9: 8.0 - 9.0 class 5: 4.0 - 5.0 class 10: 9.0 - 10.0
- 16. • We assume that a measurement that falls on the border between two intervals belongs to the previous interval (e.g. the value 4.0 belongs to class 4 instead of class 5). • By counting the number of observations that fall into each class, we get the following frequency distribution:
- 17. Measurements # of obj. Relative Frequency Percentage Frequency 0.0 - 1.0 3 0.075 7.5 % 1.0 - 2.0 4 0.100 10.0 % 2.0 - 3.0 4 0.100 10.0 % 3.0 - 4.0 7 0.175 17.5 % 4.0 - 5.0 6 0.150 15.0 % 5.0 - 6.0 5 0.125 12.5 % 6.0 - 7.0 5 0.125 12.5 % 7.0 - 8.0 1 0.025 2.5 % 8.0 - 9.0 2 0.050 5.0 % 9.0 - 10.0 3 0.075 7.5 % Totals 40 1.000 100.0 %
- 19. • In the interval 4.0 - 5.0 of the above frequency distribution, 4.0 is the lower class limit, while 5.0 is the upper class limit (Actually, 4.0 belongs to the previous interval, but since any measurement just slightly greater than 4.0 falls into the 4.0 - 5.0 class, 4.0 is essentially the lower limit of the interval). • The class width is 5.0 - 4.0 = 1.0; in fact, the class widths of all the intervals are 1.0, as is preferred.
- 20. Measurements Cumulative Frequency Relative Cumulative Frequency Percentage Cumulative Frequency 0.0 - 1.0 3 0.075 7.5 % 1.0 - 2.0 7 0.175 17.5 % 2.0 - 3.0 11 0.275 27.5 % 3.0 - 4.0 18 0.450 45.0 % 4.0 - 5.0 24 0.600 60.0 % 5.0 - 6.0 29 0.725 72.5 % 6.0 - 7.0 34 0.850 85.0 % 7.0 - 8.0 35 0.875 87.5 % 8.0 - 9.0 37 0.925 92.5 % 9.0 - 10.0 40 1.000 100.0 % Totals 40 1.000 100.0 % The following table shows the cumulative frequency distribution for the above set of measurements:
- 21. • Frequency distribution tables can also be utilized for qualitative data. • Since qualitative data is not ordinal (e.g. can be ordered), the concept of cumulative frequency does not apply when dealing with qualitative data.
- 22. Favorite Color # of persons Relative Frequency Percentage Frequency RED 16 0.2000 20.00 % BLUE 18 0.2250 22.50 % YELLOW 10 0.1250 12.50 % GREEN 9 0.1125 11.25 % ORANGE 13 0.1625 16.25 % BLACK 6 0.0750 7.50 % others 8 0.1000 10.00 % Totals 80 1.0000 100.00 % EX. The following table illustrates the use of frequency distributions with qualitative data.
- 23. Thank You