Tables & Relative Frequency (College Board AP® Statistics)

Frequency tables

How are frequency tables used for ungrouped data?

• Frequency tables can be used for ungrouped data when you have lots of the same values within a data set

  • They can be used to collect and present data easily

• If a particular value has a frequency of 3 this means that there are three of that value in the data set

• For example, the number of pets owned by a group of individuals

  • could be presented as a list, e.g. 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3 , 3, 3, 3, 3

  • or alternatively, in a frequency table

How are measures of center calculated from frequency tables with ungrouped data?

• The mode is the data value that has the highest frequency

• The median is the middle value of the data when put in order

  • Use cumulative frequencies (running totals) to find the median

    • The median is the data value that is halfway through the total frequency

• The mean can be calculated by

  • multiplying each value by its frequency 

  • summing these together to get

  • then dividing by the total frequency  = Σf_i

  • The formula, , is not given in the exam

• Your calculator can calculate these statistical measures by inputting the values and their frequencies into your calculator and calculating one-variable statistics

How are measures of variability calculated from frequency tables with ungrouped data?

• The range is the largest value of the data minus the smallest value of the data

  • It is not the largest frequency minus the smallest frequency

• The interquartile range, IQR, is the third quartile subtract the first quartile,

  • The quartiles can be found by listing out the data and calculating by hand

    • or by using that Q1 is the data value that is a quarter of the way through the total frequency etc.

  • or by inputting the values and their frequencies into your calculator and calculating one-variable statistics

• The standard deviation and variance can also be calculated by hand using the formulas after listing out the values

  • or by inputting the values and their frequencies into your calculator and calculating one-variable statistics

How are frequency tables used for grouped data?

• Frequency tables can be used for grouped data when you have lots of values within the same interval

  • Class intervals will be written using inequalities and without gaps

    • and 

• If a particular class interval has a frequency of 3 this means that there are three data items in that class interval

  • You do not know the exact data values when you are given grouped data

• For example, the heights of students within a class

  • could be presented in a grouped frequency table

Relative frequency tables

What is a relative frequency table for ungrouped data?

• A relative frequency table gives the proportion of data items falling into each category

  • This may be presented as a decimal, fraction or percentage

• If a particular value has a relative frequency of 0.4 this means that 0.4 or 40% of the items in the data set have that value

  • We do not know the exact number of data items in each group

  • However, we can calculate this if we are also told the total frequency

• For example, the ungrouped relative frequency table below shows the number of siblings that a particular group of individuals have

  • 30% of the group have no siblings

  • 50% of the group have 1 sibling

  • 15% of the group have 2 siblings

  • 5% of the group have 3 siblings

What is a relative frequency table for grouped data?

• The same idea of relative frequency can be applied to grouped data

  • If a particular class interval has a relative frequency of this means that or 70% of the items in the data set have a value within that class interval

    • We do not know the exact number of data items in each group

    • However, we can calculate this if we are also told the total frequency

• For example, the grouped relative frequency table below shows the number of weights of a sample of cats

  • 8% of the group weigh between 1 kg and 2 kg

  • 42% of the group weigh between 2 kg and 3 kg

  • 31% of the group weigh between 3 kg and 4 kg

  • 19% of the group weigh between 4 kg and 5 kg