# Using statistical measures

This resource list is designed to provide opportunity for students to explore the appropriate use of measures of central tendency and measures of spread and apply statistics to describe a population.

Visit the secondary mathematics webpage to access all lists.

### Understanding Mean, Median, Mode and Range S4

This resource is designed to enable students to understand the terms: mean, median, mode, range and to explore the relationships between these measures and their relationship to the shape of a distribution. Most students will have met the terms mean, median, mode and range but they may not have a clear understanding of their meaning and the relationships between them. One purpose of this session is to expose and discuss any misconceptions.

An applet is available to explore existing knowledge and possibly expose some initial misconceptions. A card sort activity is used to further challenge students’ understanding.

These materials exemplify the ideas and approaches adopted in the Standards Unit pilot. To get best value from them use them in association with the guidance and other materials in Improving Learning in Mathematics: Challenges and Strategies.

### Frogs

In this lesson plan and associated presentation students are required to remember how many frogs are on a slide when they have only had a few seconds to view the situation. A student who has not been present is then asked to question the students about how many frogs were on the slide. Students find simple means, modes and medians, calculate statistics from discrete and group data. Students are also required to interpret the data. Students also consider how the sample size affects the data.

### Happy Planet Index

The balance between economic success and the environment is the focus of this resource providing an example of how mathematics can help to make people more aware of important issues. Students apply statistical measures to interpret a variety of data which is then summarised in a Happy Planet Index.

A detailed lesson plan and data sheets support the implementation of this activity, which is designed to prompt discussion and develop mathematical thinking.

### Descriptive Statistics

This resource is designed to help students understand different measures of average and different measures of spread. Students also compare statistical diagrams and their use.

**Mean and Standard Deviation**: Students are given five numbers. The mean, variance and standard deviation have been calculated, showing each step of the calculations. Work card 1.1 leads students through exercises which explore the effect on the mean and standard deviation of changing the original five numbers.

**Mean and Median**: A frequency distribution of students' examination marks is given for which the mean and median have been calculated and displayed on a histogram. Work card 1.2 asks students to explore the effect on the value of the mean, the median and the shape of the distribution as the frequencies are changed.

**Standard Deviation and inter Quartile range**: A frequency distribution of students' examination marks is given for which the standard deviation and interquartile range have been calculated and displayed on a histogram. Work card 1.3 asks students to explore the effect on the value of the standard deviation and the interquartile range as the frequencies are changed.

**Boxplots**: A data set of 48 numbers is given. The median, quartiles, interquartile range and fences have been calculated and the boxplot and outliers plotted. Work card 1.4 explores the effect on the boxplots of altering the original data.

**Histograms**: A data set of 48 numbers is given which is tallied into seven classes of unequal width. The resulting histogram is displayed. Work card 1.5 explores how the histogram changes as the data values and class intervals are altered.

### Statistics in Your World - Level 4

The ‘Retail Price Index’ unit introduces students to the concept of an index number. Students should be able to calculate a price relative, (called index number in the student material) for an individual item and for a number of un-weighted items; calculate weights using frequency of purchase and use the weights to calculate a weighted price index. Students also calculate a retail price index given individual indices, or values in the base year and the current year. Students practice completing tables from home, and optionally from shops in a survey of local prices and calculating an arithmetic mean.

The ‘Equal Pay’ unit investigates whether the Equal Pay Act is working, by applying statistical techniques, and considers some of the inherent problems in making comparisons. The median, cumulative percentages and the interquartile range are covered.

For each unit there are comprehensive teacher's notes giving an overview of the unit, the aims and objectives for that unit, and prior learning or prerequisites and the equipment required.

### Interpreting Data

These resources are aimed at mathematics practitioners. They include objectives and a description of the activity. The activities highlight possible misconceptions and suggestions for how the activities might be extended.

The sessions in this resource include

- Misleading statistics (exposing and discussing common misconceptions)
- Using moving averages to identify trends (using rich collaborative tasks)

### Comparing Datasets

This activity uses Geogebra to compare different data sets. The activity ‘Comparing Datasets’ provides examples and ideas for comparing data using mean, mode median, interquartile range and standard deviation. The examples suggested are:

- Do males have faster reaction times than females?
- Is life expectancy higher in the North or South of England?
- Is X Factor more popular than Strictly Come Dancing?
- Who played the better football in the 2011 Champions League final?

There are brief suggestions for how to plan this activity, including aims and ideas for support and challenge. Before doing this activity students should know what the mean and median are. Familiarity with standard deviation and quartiles will be helpful; however this activity can be used to introduce these concepts. It will be useful if they have used Geogebra before and they know how to copy and paste data from tables. After doing this activity students can consolidate the learning by practising the techniques of finding mean/median and drawing statistical diagrams.

### Data Sets: Calculating an Estimate of the Mean

The first activity in this interactive excel file shows the number of passengers per flight from London to Paris. The raw data is shown in a grouped frequency table and the midpoint of each group can be revealed as well as each stage in the calculation of the estimate of the mean. The next two activities each show two sets of data which are dealt with in the same manner so that the calculated estimates of the mean can be compared. There are seven more sheets of questions which may be suitable for use in the classroom.

### Navigating numbers: the ONS data education programme

This resource is designed to help students understand how and why the ONS uses the public’s data to produce statistics. It includes a series of five classroom toolkits centred around an Office for National Statistics (ONS) tool or dataset that relates to a part of everyday life, including the gender pay gap; inflation; the census and understanding population dynamics; health and well-being and business, industry and trade.