AP®StatisticsCollege BoardStudy GuidesUnit 1: Exploring One-Variable DataSummary StatisticsTables & Relative Frequency

Tables & Relative Frequency (College Board AP® Statistics) : Study Guide

Naomi C

Written by: Naomi C

Reviewed by: Dan Finlay

Updated on

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

Number of pets

0

1

2

3

Frequency

3

5

4

6

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 x subscript i by its frequency f subscript i

    • summing these together to get sum for blank of f subscript i x subscript i

    • then dividing by the total frequency n equals sum for blank of f subscript i = Σfi

    • The formula, x with bar on top equals fraction numerator sum from i equals 1 to n of f subscript i x subscript i over denominator n end fraction, 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, IQR equals straight Q 3 minus straight Q 1

    • 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

      • 10 less or equal than x less than 20 and 20 less or equal than x less than 30

  • 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

Height, h

Frequency

150 less or equal than h less than 155

1

155 less or equal than h less than 160

3

160 less or equal than h less than 165

5

165 less or equal than h less than 170

4

170 less or equal than h less than 175

2

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

Number of siblings

0

1

2

3

Relative frequency

0.3

0.5

0.15

0.05

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 7 over 10 this means that 7 over 10 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

Weight of cat, w (kg)

1 less than w less or equal than 2

2 less than w less or equal than 3

3 less than w less or equal than 4

4 less than w less or equal than 5

Relative frequency

8 over 100

42 over 100

31 over 100

19 over 100

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Author
Reviewer
Naomi C

Author: Naomi C

Expertise: Maths Content Creator

Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.

Dan Finlay

Reviewer: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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