Tables & Relative Frequency (College Board AP® Statistics)

Revision Note

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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

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 $x_{i}$ by its frequency $f_{i}$
- summing these together to get $\sum_{} f_{i} x_{i}$
- then dividing by the total frequency $n = \sum_{} f_{i}$ = Σf_i
- The formula, $\bar{x} = \frac{\sum_{i = 1}^{n} f_{i} x_{i}}{n}$ , 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

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, $IQR = Q 3 - Q 1$
- 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

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
  - $10 \leq x < 20$ and $20 \leq x < 30$
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

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

The same idea of relative frequency can be applied to grouped data
- If a particular class interval has a relative frequency of $\frac{7}{10}$ this means that $\frac{7}{10}$ 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)	$1 < w \leq 2$	$2 < w \leq 3$	$3 < w \leq 4$	$4 < w \leq 5$
Relative frequency	$\frac{8}{100}$	$\frac{42}{100}$	$\frac{31}{100}$	$\frac{19}{100}$

the (exam) results speak for themselves:

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