{"id":29,"date":"2021-01-12T22:19:24","date_gmt":"2021-01-12T22:19:24","guid":{"rendered":"https:\/\/textbooks.jaykesler.net\/introstats\/chapter\/frequency-distributions\/"},"modified":"2024-08-27T20:01:06","modified_gmt":"2024-08-27T20:01:06","slug":"frequency-distributions","status":"publish","type":"chapter","link":"https:\/\/textbooks.jaykesler.net\/introstats\/chapter\/frequency-distributions\/","title":{"rendered":"Frequency Distributions"},"content":{"raw":"\r\n[latexpage]\r\n<\/span>\r\n

Frequency<\/h3>\r\n

T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>wenty students were asked how many hours they worked per day. Their responses, in hours, are as follows: 5; <\/span><\/span>6; <\/span><\/span>3; <\/span><\/span>3; <\/span><\/span>2; <\/span><\/span>4; <\/span><\/span>7; <\/span><\/span>5; <\/span><\/span>2; <\/span><\/span>3; <\/span><\/span>5; <\/span><\/span>6; <\/span><\/span>5; <\/span><\/span>4; <\/span><\/span>4; <\/span><\/span>3; <\/span><\/span>5; <\/span><\/span>2; <\/span><\/span>5; <\/span><\/span>3<\/span><\/span>.<\/p>\r\n

Table 1.9<\/a> lists the different data values in ascending order and their frequencies.<\/p>\r\n\r\n

\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
DATA VALUE<\/th>\r\nFREQUENCY<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
2<\/span><\/td>\r\n3<\/td>\r\n<\/tr>\r\n
3<\/span><\/td>\r\n5<\/td>\r\n<\/tr>\r\n
4<\/span><\/td>\r\n3<\/td>\r\n<\/tr>\r\n
5<\/span><\/td>\r\n6<\/td>\r\n<\/tr>\r\n
6<\/span><\/td>\r\n2<\/td>\r\n<\/tr>\r\n
7<\/span><\/td>\r\n1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
Table <\/span>1.9<\/span> Frequency Table of Student Work Hours<\/span><\/div>\r\n<\/div>\r\n

A frequency<\/span> is the number of times a value of the data occurs. According to Table 1.9<\/a>, there are three students who work two hours, five students who work three hours, and so on. The sum of the values in the frequency column, 20, represents the total number of students included in the sample.<\/p>\r\nA frequency distribution<\/strong> (also known as a grouped frequency distribution table<\/strong> or GFDT <\/strong>for short) is the table that lists the class<\/strong> (this is the \"Data Value\" column in Table 1.9) and the frequency. Each class can be a single value like in Table 1.9, or a range of values like in Table 1.12 below.\r\n

A relative frequency<\/span> is the ratio (fraction or proportion) of the number of times a value of the data occurs in the set of all outcomes to the total number of outcomes. To find the relative frequencies, divide each frequency by the total number of students in the sample\u2013in this case, 20. Relative frequencies can be written as fractions, percents, or decimals.<\/p>\r\n\r\n

\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
DATA VALUE<\/th>\r\nFREQUENCY<\/th>\r\nRELATIVE FREQUENCY<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
2<\/span><\/td>\r\n3<\/td>\r\n$\\frac{3}{20}$ or 0.15<\/td>\r\n<\/tr>\r\n
3<\/span><\/td>\r\n5<\/td>\r\n$\\frac{5}{20}$ <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>or 0.25<\/td>\r\n<\/tr>\r\n
4<\/span><\/td>\r\n3<\/td>\r\n$\\frac{3}{20}$ or 0.15<\/td>\r\n<\/tr>\r\n
5<\/span><\/td>\r\n6<\/td>\r\n$\\frac{6}{20}$ or 0.30<\/td>\r\n<\/tr>\r\n
6<\/span><\/td>\r\n2<\/td>\r\n$\\frac{2}{20}$ or 0.10<\/td>\r\n<\/tr>\r\n
7<\/span><\/td>\r\n1<\/td>\r\n$\\frac{1}{20}$ or 0.05<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
Table <\/span>1.10<\/span> Frequency Table of Student Work Hours with Relative Frequencies<\/span><\/div>\r\n<\/div>\r\n

The sum of the values in the relative frequency column of Table 1.10 is $\\frac{20}{20}$<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> , or 1.<\/p>\r\n

Cumulative frequency<\/span> is the accumulation of the previous frequencies. To find the cumulative frequencies, add all the previous frequencies to the frequency for the current row, as shown in Table 1.11a.<\/p>\r\n\r\n

\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
DATA VALUE<\/th>\r\nFREQUENCY<\/th>\r\nCUMULATIVE RELATIVE \r\n<\/span>FREQUENCY<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
2<\/span><\/td>\r\n3<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
3<\/span><\/td>\r\n5<\/td>\r\n8<\/td>\r\n<\/tr>\r\n
4<\/span><\/td>\r\n3<\/td>\r\n11<\/td>\r\n<\/tr>\r\n
5<\/span><\/td>\r\n6<\/td>\r\n17<\/td>\r\n<\/tr>\r\n
6<\/span><\/td>\r\n2<\/td>\r\n19<\/td>\r\n<\/tr>\r\n
7<\/span><\/td>\r\n1<\/td>\r\n20<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
Table <\/span>1.11a<\/span> Frequency Table of Student Work Hours with Relative and Cumulative Relative Frequencies<\/span><\/div>\r\n<\/div>\r\n

The last entry of the cumulative relative frequency column is one, indicating that one hundred percent of the data has been accumulated.<\/p>\r\n

Cumulative relative frequency<\/span> 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, as shown in Table 1.11b.<\/p>\r\n\r\n

\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
DATA VALUE<\/th>\r\nFREQUENCY<\/th>\r\nRELATIVE \r\n<\/span>FREQUENCY<\/th>\r\nCUMULATIVE RELATIVE \r\n<\/span>FREQUENCY<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
2<\/span><\/td>\r\n3<\/td>\r\n$\\frac{3}{20}$ or 0.15<\/td>\r\n0.15<\/td>\r\n<\/tr>\r\n
3<\/span><\/td>\r\n5<\/td>\r\n$\\frac{5}{20}$ <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>or 0.25<\/td>\r\n0.15 + 0.25 = 0.40<\/td>\r\n<\/tr>\r\n
4<\/span><\/td>\r\n3<\/td>\r\n$\\frac{3}{20}$ or 0.15<\/td>\r\n0.40 + 0.15 = 0.55<\/td>\r\n<\/tr>\r\n
5<\/span><\/td>\r\n6<\/td>\r\n$\\frac{6}{20}$ or 0.30<\/td>\r\n0.55 + 0.30 = 0.85<\/td>\r\n<\/tr>\r\n
6<\/span><\/td>\r\n2<\/td>\r\n$\\frac{2}{20}$ or 0.10<\/td>\r\n0.85 + 0.10 = 0.95<\/td>\r\n<\/tr>\r\n
7<\/span><\/td>\r\n1<\/td>\r\n$\\frac{1}{20}$ or 0.05<\/td>\r\n0.95 + 0.05 = 1.00<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
Table <\/span>1.11b<\/span> Frequency Table of Student Work Hours with Relative and Cumulative Relative Frequencies<\/span><\/div>\r\n<\/div>\r\n

The last entry of the cumulative relative frequency column is one, indicating that one hundred percent of the data has been accumulated.<\/p>\r\n\r\n

\r\n

NOTE<\/span><\/h3>\r\n<\/header>
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Because of rounding, the relative frequency column may not always sum to one, and the last entry in the cumulative relative frequency column may not be one. However, they each should be close to one.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n

Example - Professional Soccer Players<\/h3>\r\n

Table 1.12<\/a> represents the heights, in inches, of a sample of 100 male semiprofessional soccer players.<\/p>\r\n\r\n

\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
HEIGHTS \r\n<\/span>(INCHES)<\/th>\r\nFREQUENCY<\/th>\r\nRELATIVE \r\n<\/span>FREQUENCY<\/th>\r\nCUMULATIVE FREQUENCY<\/th>\r\nCUMULATIVE \r\n<\/span>RELATIVE \r\n<\/span>FREQUENCY<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
60\u201361<\/span><\/td>\r\n5<\/td>\r\n$\\frac{5}{100}$ = 0.05<\/td>\r\n5<\/td>\r\n0.05<\/td>\r\n<\/tr>\r\n
62\u201363<\/span><\/td>\r\n3<\/td>\r\n$\\frac{3}{100}$ = 0.03<\/td>\r\n8<\/td>\r\n0.05 + 0.03 = 0.08<\/td>\r\n<\/tr>\r\n
64\u201365<\/span><\/td>\r\n15<\/td>\r\n$\\frac{15}{100}$ = 0.15<\/td>\r\n23<\/td>\r\n0.08 + 0.15 = 0.23<\/td>\r\n<\/tr>\r\n
66\u201367<\/span><\/td>\r\n40<\/td>\r\n$\\frac{40}{100}$ = 0.40<\/td>\r\n63<\/td>\r\n0.23 + 0.40 = 0.63<\/td>\r\n<\/tr>\r\n
68\u201369<\/span><\/td>\r\n17<\/td>\r\n$\\frac{17}{100}$ = 0.17<\/td>\r\n80<\/td>\r\n0.63 + 0.17 = 0.80<\/td>\r\n<\/tr>\r\n
70\u201371<\/span><\/td>\r\n12<\/td>\r\n$\\frac{12}{100}$ = 0.12<\/td>\r\n92<\/td>\r\n0.80 + 0.12 = 0.92<\/td>\r\n<\/tr>\r\n
72\u201373<\/span><\/td>\r\n7<\/td>\r\n$\\frac{7}{100}$ = 0.07<\/td>\r\n99<\/td>\r\n0.92 + 0.07 = 0.99<\/td>\r\n<\/tr>\r\n
74\u201375<\/span><\/td>\r\n1<\/td>\r\n$\\frac{1}{100}$ = 0.01<\/td>\r\n100<\/td>\r\n0.99 + 0.01 = 1.00<\/td>\r\n<\/tr>\r\n
<\/td>\r\nTotal = 100<\/strong><\/td>\r\nTotal = 1.00<\/strong><\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
Table <\/span>1.12<\/span> Frequency Table of Soccer Player Height<\/span><\/div>\r\n<\/div>\r\n

The data in this table have been grouped<\/strong> into the following intervals, which are the classes<\/span>:<\/p>\r\n\r\n