For example, if you wanted to find out "What times do students get up in the morning?" you would need to clarify:

• Is it a normal school day, a weekend or a holiday e.g. “What time do students get up on a normal school day?”
• Will it matter if students have different school starting times? Compare the question “How long before school starts do students get up?”
• What units do you want to use to collect the data? “How many minutes before school starts do students get up?”
• How will part units be reported? Do you want data to the closest whole number? If you plan to have fractions can these be decimals?

• ‘student’ is defined as Year 7 and Year 11 Australian school students
• ‘getting up time’ is the time students get out of bed on a school day.

Participant characteristics
Next you need to specify the scope of your sample. For example, are you looking at a particular age group or year level or location? In these cases. your question might be “How many minutes before school starts do Year 7 students get up compared with Year 11 students?” or "How many minutes before school starts do students in Queensland get up compared with students in South Australia?".

Sample size
Estimates are made about the total population and subgroups based on the information from the sample. Generally, larger samples will give a more accurate representation of the population. However, it can be difficult to obtain accurate information on smaller groups within the population if the sample size is small.
In addition, the level of accuracy can usually be measured. There are formulae to determine the size of the sample that should be taken depending on the level of confidence required. One of the simplest is:
Sample size = √n
(where n is the size of the population)

Randomness
To allow predictions to be confidently made about the total population, samples need to be randomly selected as well as of sufficient size. For data to be selected randomly, each data item must have the same chance of being selected as any other. Pulling data items from a hat or using the random number generator on a calculator are common ways of ensuring that data are selected randomly. Data not selected randomly may be biased towards a particular outcome.

Types of Sampling
There are a number of ways that a sample can be randomly drawn from a population. For example, you may want to ensure that each subgroup of a population is represented in the same proportion as in the general population.
For more information on types of sampling see our Glossary page.

Interview
In an interview, a participant can ask questions if they haven’t understood something.

Written response
Written responses can be completed by many participants at the same time and are quicker than interviews.

Variables
A variable is any measurable characteristic or attribute that can have different values for different subjects – for example, eye colour, distance from school etc.
Characteristic is another way of saying variable. For example, height, age or country of birth are all characteristics or variables of people.

Frequency tables
A useful way to record raw data is a tally table or frequency table.
A frequency table counts the number of times – or frequency – a value occurs in the data. For example, twenty people are asked "How many TVs do you have in your household?" If 2 households have 1 TV, the frequency of households with 1 TV is 2.

Frequency tables with class intervals
When a variable has a large spread, the values can be grouped together to make the data easier to manage and present.
For example, if you asked students how much time it takes them to get to school each day, their responses may vary considerably. In this case, you can group the responses together in 5 minute intervals. These intervals are called class intervals. All class intervals should have an equal range. Class intervals are usually in groups of 5, 10, 20, 50 etc.

A graph should:

• be simple and not too cluttered
• show data without changing the data’s meaning
• show any trend or differences in the data
• be accurate in a visual sense – for example, if one value is 15 and another 30, then 30 should be twice the size of 15

• avoiding 3D representations
• avoiding broken or uneven scales

Bar graphs – Figure 3
A bar graph is used to represent categorical or discrete numerical data. A bar graph has a gap between each bar or set of bars and the widths of the gaps and bars are consistent. The length of the bars in a bar graph is also important: the greater the length, the greater the value. A bar graph can be either horizontal or vertical – vertical bar graphs are also known as column graphs.

Horizontal bar graphs – Figure 4
The advantage of using a horizontal bar graph over a column graph is that the category labels in a horizontal bar graph can be fully displayed making the graph easier to read.

Side by side bar or column graphs – Figure 5
Side by side bar graphs are useful to compare two or more groups for the same data.

Stacked bar or column graphs – Figure 6
Stacked or segmented bar graphs are sometimes used to display a breakdown of data for a particular group – for example, the breakdown of types of internet use by sex. They are most useful when the data is shown as a percentage value and the number of categories is very small. If there are too many categories, the frequency of the datasets can be difficult to read and compare.

Histograms – Figure 7
A histogram is similar to a column graph, however, there are no gaps between columns. Histograms are used for numerical data only. Data can be either discrete or continuous and grouped or ungrouped. A histogram should have equal width columns when possible.

Line Graphs
Line graphs are used for numerical data. They show how one variable changes in relation to another. The independent variable is always displayed on the horizontal axis. It is important to use a consistent scale on each axis so you can get an accurate sense of the data.

Time Series – Figure 8
Time series graphs are line graphs that display a trend or a pattern in data over time. The slope can be upward or negative. Patterns can be seasonal (over a short period), or cyclic (over a longer period). Alternately, a time series can show a random variation.
To use a times series to make predictions, you can smooth fluctuations by using a suitable smoothing process such as deseasonalising the data.

Pie Charts – Figure 9
Pie charts, often called pie graphs, sector graphs or sector charts, show how much each sector contributes to the total. Pie charts are useful when there are only a few categories as more than five categories can make them difficult to read.
It is very important to label each sector with its value to make comparison easier.

Dot Charts – Figure 10
A dot chart can convey a lot of information in a simple, uncluttered way. Each dot can represent one or many.

Box and Whisker Plots – Figure 11
Box and whisker plots display 5 figure summary statistics: minimum, quartile 1, median, quartile 3 and maximum for a set of numerical data.
Box and whisker plots are useful for looking at the shape of a data set. In particular, parallel box and whisker plots are used to compare two data sets and are drawn on the same number line. To help identify possible outliers, 'fences' can be drawn at 1.5 x, the IQR above Q₃ and below Q₁. For more information, see summarising Data, Measures of Spread.

Stem and Leaf Plots – Figure 12
Stem and leaf plots are an efficient way of recording numerical data because numbers in the stem apply to all values in the leaves. They are also very useful to show the shape of a distribution and, since they order data, can be used to identify median and quartiles.
Back to back stem and leaf plots are used to compare the distribution of two data sets.

Age Pyramids – Figure 13
Age pyramids are used to represent a population age structure. Age pyramids are a very effective way of showing change in a country’s age structure over time or for comparing different countries. Estimates and projections of Australia's population from 1971 to 2050 are available on the ABS Animated Age Pyramid page.

Figure 4: A horizontal bar graph

Figure 5: A side by side column graph

Figure 6: A stacked bar graph

Figure 7: A histogram showing data with grouped intervals

Figure 8: A time series graph


Figure 9: A pie chart

Figure 10: A dot plot with many to one correspondence


Figure 11: Parallel box and whisker plots

Figure 12: A back to back stem and leaf plot of arm span


Figure 13: An age pyramid (Source: Australian Bureau of Statistics Education Services)

• Will you find the mean or median or both?
• Will you exclude extreme values?

a) Measures of centre
The centre of a set of data is important. Often you want to know which value occurs most commonly or the average of a set of values.

Numerical data
Mean: The arithmetic mean is calculated by finding the sum of all values in the data set and then dividing it by the number of values in the set. The mean is often called the average.
Median: The median is the middle value of a dataset when all the values are arranged from smallest to largest number. If a dataset contains an even number of values, the median is taken as the mean of the middle two. The median is the preferred measure of centre when the data is not symmetric or contains outliers.

Numerical and categorical data – Figure 14
Mode: The mode is the value that occurs most often in a data set.

b) Measures of Spread
The spread of a data set, when combined with its centre, will give a more complete picture.

Range
The range is the distance from the minimum value to the maximum value in the data set.
Range = maximum – minimum

Interquartile range – Figure 15
Quartiles: As the name suggests, quartiles divide data into four equal sets.
When observations are placed in ascending order according to their value, the first or lower quartile is the value of the observation at or below which one-quarter (25%) of observations lie. The second quartile is the median at or below which half (50%) of observations lie. The third or upper quartile is the value of the observation at or below which three-quarters (75%) of the observations lie.
Another way to think about this is: the median divides the data into two equal sets: the lower quartile is the value of the middle of the lower half, and the upper quartile is the value of the middle of the upper half. The difference between upper and lower quartiles (Q₃ - Q₁) also indicates the spread of a data set. This is called the interquartile range (IQR).

Interquartile Range = Upper quartile - Lower quartile
The interquartile range spans 50% of a data set, and eliminates the influence of outliers because, in effect, the highest and lowest quartiles are removed.
Calculating the IQR will help you to identify potential outliers in the data. Any data above Q₃ +1.5 x IQR (upper fence) and below Q₁ – 1.5 x IQR (lower fence) should be investigated to decide whether or not these observations need to be excluded from the data before it is analysed further.

Outliers
A potential outlier might be a mistake or an extreme value so you need to check the original data to determine if it is to be discarded or retained. However, cleaning data by discarding extreme values might give you a false view of variations that can be found in datasets. In turn, this could result in inaccurate modelling based on the data and false conclusions.

The standard deviation – Figure 16
The standard deviation measures the average distance each value in a data set is from the mean. A data set with a higher standard deviation will be more spread out than one with a lower standard deviation. Compare the following two data sets:

c) Shape
The standard distribution – Figure 17
Data which has a standard distribution is characterised by a symmetric, single curve. Also known as the normal or bell shaped distribution.
You would expect data such as the height of Year 9 students or the length of cane toads to show a standard distribution. A standard distribution results from most of the data clustering around the mean and less data occurring further away from the mean.

Bi modal – Figure 18
Bi modal data usually comes from two distinct populations. The bar chart of Year 5 girls' and Year 10 boys' heights shows two distinct modal groups.

Skew – Figure 19
Some non-symmetric data distributions can be described as skewed.
Data which is positively skewed typically has a cluster of lower end values and a taper of higher end values. Data which is negatively skewed typically has a cluster of higher end values and a taper of lower end values.
You could expect data such as the amount of total weekly income to show a negative skew. The number of people on a higher weekly income is fewer than those on low or medium incomes. Data you could expect to show a positive skew might be the amount of hours spent on Facebook by age. It's likely that the amount of time spent on Facebook will peak in the late teens to twenties and then taper off as a person's age increases.

Figure 19: Bi modal graph of distribution of height


Figure 20: Positive skew, distribution of age of Indigenous males

Accuracy of Data
All your calculations need to be accurate, verifiable from the data and clearly communicated using simple language.

Misinterpretation
To effectively communicate your results, you will need to be aware of avoiding any misinterpretation of the data such as using the mean when the median is more appropriate or not taking seasonal variation into account.

Stating your conclusions
With statistics, there is always a risk that the results you have do not tell the whole story. You can use the following checklist to help judge the reliability of your statistical information.

• Do your conclusions communicate the message told by the data?
• Are your conclusions based on results rather than on your opinions?
• Have you considered alternative explanations for the same results?
• Is your report set out logically including using an organisational framework such as headings and sub headings?
• Have you included the source of any information you have used or referred to?
• Have you included relevant tables and graphs?
• Are your findings clear, related to your aim and only contain necessary information?

• Have you considered your audience and used appropriate language?
• Have you anticipated questions your reader might have? For example, have you explained unusual or unexpected results? Have you justified your choice of analysis, indicated your sampling process etc?
• Can your reader check your conclusions by viewing your analysis?