Statistics Toolkit (Edexcel IGCSE Maths A)

To find the new mean of a data set if a new data item is added to the set:

1. Find the total of the current data set:

2. Add the new data item

3. Divide by the new total number of data items

E.g. for a data set of 12 items and a mean of 5.7, the total of the values is .
Therefore when a new data item of 9.6 is added to the set, the new mean is .

To find the mean from a frequency table:

1. Include a column for (value frequency).

2. Add the values in this column.

3. Divide the sum by the total frequency.

To find the median from a frequency table:

1. Make sure the values in the table are in order.

2. Find the th value, where is the total frequency.

To find the mode from a frequency table, look for the value with the highest frequency.

To estimate the mean from grouped data:

1. Find the midpoint of each class.

2. Calculate (midpoint × frequency) for each class.

3. Sum the (midpoint × frequency) values.

4. Divide the sum by the total frequency.

A class interval is a range of values within which data points are grouped together.

False.

Only an estimate of the mean can be calculated from grouped data.

To find the class interval containing the median from grouped data:

1. Find the position of the median using , where is the total frequency.

2. Use the table to determine the class interval containing this position.

To find the modal class interval from grouped data, look for the class interval with the highest frequency.

The phrase "estimate the mean" usually indicates that the data is grouped, and that you should use the midpoint method to estimate the mean.

The range is the difference between the highest and lowest values in a data set (i.e., highest value minus lowest value).

A quartile is one of the values (lower quartile, median and upper quartile) that divide an ordered data set into four equal parts.

The interquartile range (IQR) of a data set is the difference between the upper and lower quartiles (i.e., upper quartile minus lower quartile).

The lower quartile (LQ) is the value below which 25% of the data lies (and above which 75% of the data lies).

The upper quartile (UQ) is the value above which 25% of the data lies (and below which 75% of the data lies).

False.

The range is affected by outliers.

True.

The IQR is not affected by outliers.

Lower quartile th value

• where is the total frequency

Upper quartile th value

• where is the total frequency

The equation for the interquartile range (IQR) is  

Where:

• is the upper quartile

• is the lower quartile

When comparing distributions, you should compare two things:

• An average for the distributions, e.g. mean, median or mode.

• A measure of spread for the distributions, e.g. range or IQR.

False.

The mean or median are usually used to compare the averages of distributions.

The mode can be used for non-numerical data.

The range or interquartile range (IQR) can be used to compare the spread of distributions.

The IQR focuses on the middle 50% of the data.

To compare the averages:

1. Give the numerical values of the averages,
   e.g. data set A has a mean of 43.5 and data set B has a mean of 53.7.

2. Explicitly compare them,
   e.g. the mean of set B is greater than the mean of set A.

3. Explain what the comparison means in the context of the question,
   e.g. this means that, on average, the people in set B take longer on the task than the people in set A.

To compare the spreads:

1. Give the numerical values of the range or interquartile range,
   e.g. set A has a range of 7, set B has a range of 10

2. Explicitly compare them,
   e.g. the range of set B is greater than the range of set A.

3. Explain what the comparison means in the context of the question,
   e.g. this means that the times taken by individuals in set B is more spread out than the times taken by individuals in set A.

True.

When comparing raw data sets, you should check for extreme values in either distribution and mention how they may affect the reliability of the results and comparisons.

True.

You should make at least two pairs of comments when comparing distributions, one pair comparing averages and one pair comparing spread.

True.

When comparing distributions, you should also consider any assumptions or potential issues with the data as these could affect the validity of the comparisons.

E.g. when assessing the water quality of a river, the samples may have all been taken from one place so may not be representative of the whole river.

False.

When comparing averages and spread, you must discuss them in the context of the question, not just compare the numbers.

The diagram shown is a pictogram.

A pictogram is a visual representation of discrete data using repeated symbols or icons.

False.

Bar charts are not used for continuous data.

Bar charts are used for discrete data.

To identify the mode from a bar chart, find the bar with the highest height or frequency.

A comparative bar chart is a bar chart that displays two or more data sets side by side for easy comparison.

False.

A key is not optional when creating a pictogram.

A pictogram requires a key that specifies the frequency represented by each symbol or icon.

True.

Bar charts should have gaps between the bars.

False.

Pictogram symbols should be of the same size for easy comparison.

(Though a pictogram may use part of a symbol, to represent a frequency that is less than the value of the complete symbol.)

True.

A two-way table is used to compare two types of characteristics.

E.g. school year group and favourite genre of movie.

1. Identify the two characteristics, e.g. favourite colours, gender

2. Use rows for one characteristic and columns for the other

3. Add an extra row and column for marginal totals

False.

When completing a two-way table, some values can be filled in directly from the question information, but some values will need to be worked out.

E.g. you may need to subtract other values in a row from the row total to find a missing value.

You can double-check your answers when completing a two-way table by making sure that all row and column totals add up correctly, and that they match the grand total.

The probability of a particular event occurring can be worked out by finding the number of successes by the total number.

E.g. the probability that a student's favourite subject is Physics is .

If a pie chart is drawn for a set of data where the total frequency is 180, you multiply the frequency of an item by 2 (i.e. 360 ÷ 180) to find the size of its angle on the pie chart.

In a pie chart, if an angle of 30° represents a frequency of 10, then you can find the total frequency by:

• dividing 10 by 30 to find how much 1° represents,

• then multiplying this by 360.

Alternatively, you can see how many times 30° goes into 360° and then multiply this by 10.

To calculate the angles needed for a pie chart:

• Divide each frequency by the total frequency.

• Multiply each result by 360°.

Alternatively:

• Divide 360° by the total frequency.

• Multiply each frequency by this number.

If you are given the angles in a pie chart and the total frequency, you can calculate the individual frequencies by doing the following:

• Divide each angle by 360°.

• Multiply by the total frequency.

Alternatively:

• Divide the total frequency by 360.

• Multiply each angle by this number.

When reading and interpreting statistical diagrams, you should look for:

• Keys

• Shading

• Axis labels

• The word "frequency"

• Any unusual or unexpected information mentioned

An anomaly, otherwise known as an extreme value or outlier, is a data point that is significantly different from the rest of the data.

True.

You may be asked to comment on aspects of a statistical diagram that could be misleading or incorrect, such as uneven gaps in axis values or a missing key.

In the context of statistical diagrams, a key is a legend that explains the meaning of symbols, colours, or shading used in the diagram.

True.

The purpose of comparing statistical diagrams is to identify and comment on differences or similarities in averages, spread, and unusual data values for the data sets represented by the diagrams.

When deciding which measures to compare in statistical diagrams, you should consider:

• Whether the mean, median or mode is the appropriate average to use.

• Whether the range or interquartile range is the appropriate measure of spread to use.

• Whether any assumptions or potential issues with the data could affect the reliability of the results and comparisons.

False.

You should aim to make at least two pairs of comments when comparing statistical diagrams:

• One pair should compare averages and comment on what this means in the context of the question.

• The other pair should compare spread and comment on what this means in the context of the question.