4.1.4 Frequency Tables

Ungrouped Data

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 the value 4 has a frequency of 3 this means that there are three 4’s in the data set

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

The mode is the value that has the highest frequency
The median is the middle value
- Use cumulative frequencies (running totals) to find the median
The mean can be calculated by
- Multiplying each value x_iby its frequency f_i
- Summing to get Σf_ix_i
- Dividing by the total frequency n = Σf_i
- This is given in the formula booklet

$\bar{x} = \frac{\sum_{i = 1}^{k} f_{i} x_{i}}{n}$

- Your GDC can calculate these statistical measures if you input the values and their frequencies using the statistics mode

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

The range is the largest value of the data minus the smallest value of the data
The interquartile range is calculated by

$IQR = Q_{3} - Q_{1}$

- The quartiles can be found by using your GDC and inputting the values and their frequencies
The standard deviation and variance can be calculated by hand using the formulae
- Variance

$σ^{2} = \frac{1}{n} \sum_{i = 1}^{k} f_{i} x_{i} ² - {\bar{x}}^{2}$

- Standard deviation

$σ = \sqrt{\frac{1}{n} \sum_{i = 1}^{k} f_{i} x_{i} ² - {\bar{x}}^{2}}$

- You do not need to learn these formulae as you will be expected to use your GDC to find the standard deviation and variance
  - You may want to see these formulae to deepen your understanding

Exam Tip

Always check whether your answers make sense when using your GDC
- The value for a measure of central tendency should be within the range of data

Worked Example

The frequency table below gives information number of pets owned by 30 students in a class.

Number of pets	0	1	2	3
Frequency	11	5	8	6

Find

the mode.

4-1-3-ib-ai-aa-sl-ungrouped-data-a-we-solution

the median.

4-1-3-ib-ai-aa-sl-ungrouped-data-b-we-solution

the mean.

4-1-3-ib-ai-aa-sl-ungrouped-data-c-we-solution

the standard deviation.

4-1-3-ib-ai-aa-sl-ungrouped-data-d-we-solution

Grouped Data

How are frequency tables used for grouped data?

Frequency tables can be used for grouped data when you have lots of the same 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 the class interval $10 \leq x < 20$ has a frequency of 3 this means there are three values in that interval
  - You do not know the exact data values when you are given grouped data

How are measures of central tendency calculated from frequency tables with grouped data?

The modal class is the class that has the highest frequency
- This is for equal class intervals only
The median is the middle value
- The exact value can not be calculated but it can be estimated by using a cumulative frequency graph
The exact mean can not be calculated as you do not have the raw data
The mean can be estimated by
- Identifying the mid-interval value (midpoint) x_i for each class
- Multiplying each value by the class frequency f_i
- Summing to get Σf_ix_i
- Dividing by the total frequency n = Σf_i
- This is given in the formula booklet

$\bar{x} = \frac{\sum_{i = 1}^{k} f_{i} x_{i}}{n}$

- Your GDC can estimate the mean if you input the mid-interval values and the class frequencies using the statistics mode

How are measures of dispersion calculated from frequency tables with grouped data?

The exact range can not be calculated as the largest and smallest values are unknown
The interquartile range can be estimated by

$IQR = Q_{3} - Q_{1}$

- Estimates of the quartiles can be found by using a cumulative frequency graph
The standard deviation and variance can be estimated using the mid-interval values x_i in the formulae
- Variance

$σ^{2} = \frac{1}{n} \sum_{i = 1}^{k} f_{i} x_{i} ² - {\bar{x}}^{2}$

- Standard deviation

$σ = \sqrt{\frac{1}{n} \sum_{i = 1}^{k} f_{i} x_{i} ² - {\bar{x}}^{2}}$

- You do not need to learn these formulae as you will be expected to use your GDC to estimate the standard deviation and variance using the mid-interval values
  - You may want to see these formulae to deepen your understanding

Exam Tip

As you can only estimate statistical measures from a grouped frequency table it is good practice to indicate that the values are not exact
- You can do this by rounding values rather than leaving as surds and fractions
- $\bar{x} = 0.333$ (3sf) rather than $\bar{x} = \frac{1}{3}$

Worked Example

The table below shows the heights in cm of a group of 25 students.

Height, $h$	Frequency
$150 \leq h < 155$	3
$155 \leq h < 160$	5
$160 \leq h < 165$	9
$165 \leq h < 170$	7
$170 \leq h < 175$	1

Write down the modal class.

4-1-3-ib-ai-aa-sl-grouped-data-a-we-solution

Write down the mid-interval value of the modal class.

4-1-3-ib-ai-aa-sl-grouped-data-b-we-solution

Calculate an estimate for the mean height.

4-1-3-ib-ai-aa-sl-grouped-data-c-we-solution

DP IB Maths: AI HL

Revision Notes

4.1.4 Frequency Tables

Ungrouped Data

How are frequency tables used for ungrouped data?

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

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

Exam Tip

Worked Example

Grouped Data

How are frequency tables used for grouped data?

How are measures of central tendency calculated from frequency tables with grouped data?

How are measures of dispersion calculated from frequency tables with grouped data?

Exam Tip

Worked Example

Author: Daniel

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