Mean, median, mode and range

You will likely see a few basic statistical concepts in QR on Test Day.

• Mean: Sum up all of the individual terms, then divide by the number of terms.
• Median: The middle term when all terms are in ascending order, or the mean of the middle two terms if you have an even number of terms.
• Mode: The most common value in the data.
• Range: The spread of the data, measured by the difference between the highest and lowest values.

These concepts may seem very familiar, given that you probably learned them at a younger age. The challenge in the UCAT is to apply the concepts quickly and accurately as you make calculations.

The UCAT is likely to include one or more twists to make it harder to apply the concept to the specific question and data that you must answer. With practice, you can improve pacing and minimise errors.

You will almost surely use the mean formula at least once on Test Day. You may need to use it, or apply the concepts involved in the formula, multiple times in the QR section.

The UCAT will normally use average as a synonym for mean. Thus, any time you see average (as a noun or verb) in a Quantitative question, consider how to apply the mean formula to work out the correct answer.

Memorise the mean formula so you can find every chance to use it as you revise.

The mean formula is a three-part formula, which means it consists of three parts with the potential for an unknown value in one part.

Most commonly in your maths education, the mean is the unknown value, so you must solve for it by applying the formula as written. You may see this in UCAT QR.

However, the UCAT loves to make you rearrange a three-part formula in order to solve for an unknown. In your UCAT revision and on Test Day, expect to use the mean formula to:

• Solve for an unknown sum when told the mean (or average) and the number of terms.
• Solve for an unknown term when told the mean (or average), the number of terms and the remaining terms that you must add up along with the unknown term.
• Solve for the number of terms when told the mean (or average) and either the sum of the terms or a list of terms that you can add up without individually counting the terms.
• Solve for a new mean (or average) based on one or more terms added or removed – so that both the sum and number of terms are different.
• Solve for an unknown term when told the old mean (or average), the new mean (or average) and the original sum and number of terms.

Watch out for varied, real-world contexts that may initially obscure the fact that you are expected to use the mean formula.

• The word ‘per’ can indicate the mean as a ratio or proportion. Example: the profit per month suggests you could add up the profit for the months in question and divide by the number of months.
• The word ‘total’ can indicate the sum of the terms involved. Example: total daily sales implies a sum of sales for individual days, so you could divide it by the number of days to find the average daily sales, or you could divide it by the average daily sales to find the number of days involved.

Expect to work with a small number of terms when you see a median question in QR. Most commonly, you will see between 3 and 6 terms; you are highly unlikely to see more than 10.

Thus, you can adjust your approach to work quickly with a handful of terms:

• Count or eyeball the number of terms. This could be as simple as a speedy visual check to see there are 3, 4, 5 or 6 terms.
• With an odd number of terms – eliminate the highest and lowest values, then repeat with remaining values until the middle term remains.
• With an even number of terms – again, eliminate the highest and lowest values, then repeat with remaining values until the two middle terms remain. Then, add the middle terms and divide by 2.
• This way, you can avoid writing out all the terms or trying to mentally resequence them. But you must work towards ascending numerical order by mentally crossing out the highest and lowest values and working towards the middle.
• This approach will become faster and more comfortable with practice.

Watch out for a mix of positive and negative values. Pay attention to the minus signs as you work, to avoid maths errors.

Watch out for QR questions that include the word median in a specialised term to describe a numerical value unique to that question or set.

Examples:

• median income
• median unit price
• median consumer satisfaction index

In such cases, you may find a term suggesting a median value without a way for you to re-order values in ascending order. For example, if the data shows that the median income in Year 1 is £35,000 and the median income in Year 5 is £37,500, you cannot solve for the median.

Instead, median is simply a data descriptor intended to make you think you must solve for the median until you see that other maths are required.

The mode is simply the most common value in the data, but the UCAT will find ways to make it difficult to find the modal value.

Examples – and possible solutions – include:

• A table with a large number of values. Make a quick note in the notebook of the two or three most frequent values, then count each carefully.
• A bar graph with many categories along the same axis. You must mentally single out the relevant bars, then look for repeated values.
• A line graph with one or more lines with repeated values. Again, look for repeated values within the question’s parameters.

Remember, you need not rewrite all the numbers to resequence them (as you might do to find the median). You simply need to check repeated numbers for the most frequent.

Watch out for wrong answer traps in questions involving the mode.

Most commonly, you may find another value that appears almost as many times as the modal value. If the modal value appears one time more than the next most common value, it is easy to make an error when working a bit too quickly.

One approach is to pair and eliminate instances of the two most frequent values.

Example: What is the modal score on Test 3?
Data: The scores on Test 3 are 87, 74, 73, 62, 73, 82, 74, 67, 81, 73, 59, 52, 74, 80, 57, 77 and 73.
At a quick glance, you might see that 73 and 74 are most frequent, so you can then mentally pair up one of each value and eliminate the pair from the list.

Proceeding in this fashion, we pair the 2nd and 3rd scores, the 5th and 7th scores, and the 10th and 13th scores. Thus, we are left with the final 73 without another 74, so 73 is the mode.

When working with the mode, you could in theory have more than one modal value.

The mode could be:

• bimodal if two values appear equally with the greatest frequency.
• multimodal if more than two values appear equally with the greatest frequency.

We would not expect to see bimodal or multimodal values on Test Day, though the UCAT could potentially include these. For example, if you must find the mode and two values both appear three times, but no other value appears more than twice, it would be accurate to say it is bimodal or – more simply – that there are two modes.

The range is the spread in one category of the data.

In the UCAT, you will normally solve for the range by subtracting the lowest value from the highest in the category.

Watch out for range questions involving negative values or a mix of positive and negative values. Remember: Subtracting a negative number is the same as adding a positive number.

Example: What is the range of midday temperatures for Week 1?
Data: The highest midday temperature is 17°C. The lowest midday temperature is –3°C.
The range is therefore 17°C – –3°C, or 17°C + 3°C = 20°C.

The UCAT may add a complication to a question involving range.

You may have to read values from a graph with confusing labels or with two y-axes, making it very easy to make an error in selecting the correct values.

You may also need to do an extra step to determine the highest and lowest values. For example, you may have to apply additional information from the question to adjust the values in the graph to come up with the correct values.