# Australian Bureau of Statistics

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 Education Services

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 Concepts and definitions

Click on the triangles to open a section. The table below is a list of the concepts covered in each section.

Statistics

Statistics are numerical data that have been organised to serve a useful purpose. A major role of the ABS is to provide the Australian community with statistics that will help them make informed decisions. Statistical information provided by the ABS is used widely in Australia by governments, business people, researchers, members of the public, teachers and students.

Data
Data are observations or facts which, when collected, organised and evaluated, become information or knowledge.

Data item
A data item is the smallest piece of information that can be obtained from a survey or census.

Dataset
A dataset is data collected for a particular study. A dataset represents a collection of elements; and for each element, information on one or more characteristics is included.

Outliers
An outlier is an extreme value of the data. It is an observation value that is significantly different from the rest of the data. There may be more than one outlier in a set of data.
Sometimes, outliers are significant pieces of data and should not be ignored. In other instances, they occur as a result of an error or misinformation and should be ignored. The decision to include or exclude an outlier needs to be clearly justified when discussing results.

Example:
The weights (in kilograms) of 30 students were measured and recorded in the stem and leaf plot shown in Figure 1. In this case, the stem is the whole number values and the leaves are the decimal values. The outliers are 56.3 and 67.7.

 Stem Leaf 56 3 57 58 4 4 9 59 0 0 2 3 8 60 0 2 4 5 7 8 9 61 1 2 4 4 5 6 7 9 9 62 1 2 3 7 63 64 65 7
Fig 1 Stem and leaf plot

Variables

 A variable is any measurable characteristic or attribute that can have different values for different subjects. Height, age, amount of income, country of birth, grades obtained at school and type of housing are examples of variables. Observation An observation is a single piece of data about a variable Independent variable An independent variable is the variable whose values are independent of changes in the values of other variables. It its the variable deliberately controlled or changed to assess changes in the dependent variable. Dependent variable A dependent variable depends on the independent variable. Categorical variables Nominal variable A nominal variable describes a name or category. For example, for the variable 'method of travel to school' all its values are words such as bus, walk, car and tram. Nominal variables are often referred to as categorical variables. Ordinal variable An ordinal variable is a number that represents a category. For example, postcodes and school year levels. Numerical variables A numerical variable is one that describes a numerically measured value. Numerical variables can be either discrete or continuous. Continuous variable A continuous variable is a numeric variable that can take any value within a certain range. For example, distance, age and temperature are continuous variables. Discrete variable A discrete variable can only take a finite number of values within a certain range. An example of a discrete variable is the number of children in a family – a family can have 0,1,2 or 3 children but not 2.5. Class interval A class interval is a group of data values for a variable. The intervals are generally the same size – for example, 4-6, 7-9 and 10-12. However, the intervals may have different sizes such as 4-6, 7-9 and 10-14. The boundaries of class intervals must not overlap so that each observation can be allocated to only one interval.

Sampling
Frequency and distribution

 The frequency (f) of a particular observation is the number of times the observation occurs in that data. Cumulative frequency Cumulative frequency is the total of a frequency and all frequencies below it in a frequency distribution. It is the running total of frequencies. Relative frequency Relative frequency is another term for proportion. It is the number of times a particular observations occurs divided by the total number of observations. Distribution The distribution of a variable is the pattern of values of the observations.

Graphs and displays
Summary statistics

Mean
The mean of a numeric variable is calculated by adding together the values of all observations in a dataset and then dividing by the number of observations in the set. It is often referred to as the average. Thus:
Mean = sum of all the observations ÷ number of observations

For example, find the mean of these numbers 5, 3, 4, 5, 7, 6.

 Mean = (5 + 3 + 4 + 5 + 7 + 6) ÷ 6 = 30 ÷ 6 = 5

Notice that the value of every member of the dataset is used to calculate the mean.

Median
The median is the middle value of a set of odd numbered data, or the mean of the middle two in an even numbered set after the data have been placed in ascending order.

For example, dataset A contains 3, 7, 1, 9, 2, 5, 9. Rearranged in ascending order it becomes: 1, 2, 3, 5, 7, 9, 9. The middle number is 5 so, the median is 5.
Dataset B contains 1, 3, 4, 5, 10, 12, 13, and also has a median of 5 although the values of the data vary considerably.

The position of the median can also be found by using the formula (n + 1) ÷ 2 , where n is the number of values in a set of ordered data.

For dataset A: n = 7

 So the position of the median = (7 + 1) ÷ 2 = 8 ÷ 2 = 4

The median is the fourth number which has a value of 5.

The above example is for an odd number of observation, i.e. n = 7. However, an extra step is necessary when the number of observations is even.
For example, if n = 8 then
 the position of the median = (8 + 1) ÷ 2 = 9 ÷ 2 = 4.5
This means that the position of the median lies between the fourth and fifth observations. To find the value of the median, add together the fourth and fifth observations and divide by two. For example, if the dataset is: 1, 1, 4, 4, 8, 9, 9, 10 then the median is, (4 + 8) / 2 = 6.

The median value is decided by its location in the ordered dataset and not because of its actual value. Notice that the values of the other members of the dataset are not taken into consideration, only their position. There are as many values above the median as there are below.

The median is usually calculated for numeric variables but may also be calculated for an ordinal nominal variable.

Mode
The mode is the most frequently observed value in a dataset. Mode is the only measure you can use when the data is categorical and has no order – for example, place of birth, favourite colour and hair colour. As the dataset is not numbers, you cannot add and divide, so you cannot find a mean. The dataset cannot be sorted from smallest to largest so you cannot find the middle value and median. The mode does not necessarily give an indication of a dataset’s centre. A set of data can have more than one mode (see Figure 1).

For example, a group friends in Year 10 have the following hair colours: red, brown, blonde, black, blonde, black, brown, brown, black, blonde, brown, brown, black.

 HAIR COLOUR FREQUENCY Red 1 1 Brown 5 5 Black 4 4 Blonde 3 3

The most common hair colour is brown so the mode is brown.

Range
The range is the actual spread of data including any outliers. It is the difference between the highest and lowest observation.
Range = maximum value – minimum value
For the following dataset of students' ages: 17, 15, 14, 16, 14, 15, 16, 12, 17, 13, 12, 17, 13, 16, 15

 Maximum value Minimum value = 17 = 12

 Range = maximum value – minimum value = 17 – 12 = 5

The range of the student's ages is 5 years.

Quartiles
Quartiles divide data into four equal groups. Using the example of 15 students above, we have the following ordered dataset: 12, 12, 13, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17. We can divide this set into four equal sized groups with each group containing one quarter of the data:

• The first quartile (Q1) is the value that 25% of the data is below.
• The second quartile (Q2) is the value that 50% of the data is below. This is the same as the median.
• The third quartile (Q3) is the value that 75% of the data is below.
 In the example: Q1 = 13 Q2 = 15 Q3 = 16

Interquartile range

The interquartile range refers to the middle 50% of data. Another way to put it is the interquartile range is the difference between the upper (75%) and lower (25%). The interquartile range is an indicator of the spread of the data. It eliminates the influence of outliers since the highest and lowest quarters are removed. The interquartile range is found by subtracting Q1 from Q3.

Five number summary (quartiles)
This is a useful way to summarise data. It consists of:

• the lowest value
• the highest value
• the first quartile (Q1)
• the third quartile (Q3)
• the second quartile (Q2).
The range can be found from the difference between the highest and lowest value. The median is the second quartile (Q2) and the interquartile range is the difference between the third and first quartiles (Q3 – Q1).

Standard deviation
Standard deviation (s) is the measure of spread most commonly used when the mean is the measure of centre. Standard deviation is most useful for symmetric distributions with no outliers.
The standard deviation for a discrete variable made up of n observations is the positive square root of the variance as show in Figure 3.

Fig 1 Unimodal, bimodal and multimodal

Fig 2 Quartiles

Fig 3 Standard deviation formula