Statistics: mean, median, mode and range
Australian students meet numerical data in ordinary places: a Woolworths or Coles receipt, a wait for a Sydney train, a trip on a Melbourne tram, or a temperature recorded in degrees Celsius. Each collection of numbers can be summarised, but the summary must fit the question being asked.
The four familiar measures describe different features of a data set. The mean balances the values, the median identifies the middle, the mode shows which value occurs most often, and the range records the span from the minimum to the maximum. Treating them as interchangeable can hide important information.
The calculations become clearer when each result stays connected to the original observations. Working through the measures one at a time also reveals why similar-looking summaries can lead to different conclusions.
What each measure tells you
The mean is the arithmetic mean, commonly called the average and one measure of central tendency. It answers a useful question about balance: if every value were shared equally, how much would each share receive? It uses every observation, which makes it sensitive to unusually high or low values.
The median describes position instead. Put the data in order and locate the middle; half the values lie on either side of it. The mode describes frequency by identifying the most common value. The range measures spread using only the two endpoints.
Before calculating anything, write down the values exactly as recorded, retain repeated observations, and check that the unit is consistent. Dollars, minutes and degrees Celsius cannot be combined into one meaningful total. For related explanations and further practice, the CurryPlum statistics page can help you check a calculation against a worked solution.
Mean and arithmetic balance
Use the formula (\bar{x}=\frac{\sum x_i}{n}), where (n) is the number of observations. For five weekly grocery totals of A$62.40, A$48.15, A$55.90, A$71.20 and A$60.35, the sum is A$298.00, so the mean is A$298.00 ÷ 5 = A$59.60.
Keeping the currency symbol and decimal places throughout makes the arithmetic easier to audit. Round only at the end if a shorter answer is required, and never round one value before adding it to the rest.
The same calculation may describe a sample or an entire population. A sample mean summarises the people or purchases included in your data; a population mean covers every case in the defined group. State which one you have rather than letting the reader guess.
Median, mode and frequency
To find the median, sort the values from smallest to largest. With an odd number of observations, take the central value; with an even number, average the two central values. The five grocery totals have a median of A$60.35. If a sixth docket of A$68.00 is included, the middle pair is A$60.35 and A$62.40, giving A$61.375, or A$61.38 to the nearest cent.
For the mode, count how often each value appears. The five original dockets have no mode because every total is different, and no value deserves the label merely by being close to the others. At a Melbourne tram stop, waits of 3, 5, 5, 8 and 10 minutes have a mode of 5 minutes.
Two modes are possible when different values tie for the highest frequency; report both. If all values occur once, there is no mode. A mode can also describe categories, such as the most common supermarket in a class survey, where averaging brand names would be meaningless.
Range, outliers and spread
The range is the largest value minus the smallest. For the original grocery totals, it is A$71.20 − A$48.15 = A$23.05. This simple span reveals how far the observations stretch, but it does not show where most values sit between the endpoints.
Consider Melbourne daily maximum temperatures of 18, 21, 19 and 24 degrees Celsius. Their range is 6 degrees Celsius. Add a fifth day at 34 degrees Celsius and the range becomes 16 degrees Celsius, even though the original four observations have not changed.
That sensitivity is why an outlier, an extreme value unlike most of the data, can dominate a range. The additional 34-degree day may be genuine, so check the source rather than deleting it. For a larger data set, the interquartile range, which covers the middle half of the ordered values, provides a more resistant measure of spread.
Choosing the right summary
Choose a measure by matching its strength to the purpose. Use the median when a typical middle value matters, the mean when balance across all observations matters, the mode for the most common case, and the range for a quick comparison of spread. Report the unit and the number of observations beside the result.
Two Sydney train waiting-time sets, 5, 8 and 11 minutes and 3, 8 and 13 minutes, both have a mean and median of 8 minutes. Their ranges are 6 and 10 minutes respectively, so the same centre conceals a different spread.
Clear presentation also depends on making the source and audience visible. This Arabic educational review offers a comparison point for educational material written for a different audience, illustrating why presentation matters. Apply the same care to numerical claims by naming the data set, the measure and the unit.
Each measure answers a slightly different question. A mean can balance a set, a median can locate its middle, a mode can reveal what recurs, and a range can expose its extent. Keeping the original list visible makes every result checkable and prevents a neat summary from claiming more than the evidence supports.
Your next step is to collect five recent supermarket dockets, record their totals in dollars and cents, and calculate the mean, median, mode and range before checking each answer against the original values.