Averages and Skewness

The average you choose is a claim

Averages look like the safest topic in statistics and they are the easiest place to mislead. Here is why this specification puts averages and skewness in the same section. If a set of data is symmetric, the mode, median and mean all land in roughly the same place and it hardly matters which you quote. As soon as the data is skewed they separate, sometimes by a lot, and quoting the mean rather than the median changes what your summary appears to say. So the question is never only what the average is. It is which average, and what that choice claims about the data.

Five kinds of average

Three you will know already, two that are particular to this specification, and the word that ties them to the rest of the section.

Average to its job

Which average fits?

A street has nine houses worth between 200 thousand and 260 thousand pounds, and one worth 4 million. Which average best represents a typical house on that street, and why?

• The median, because one extreme value barely moves it
• The arithmetic mean, because it is the only one that uses every value
• The mode, because it shows the price that occurs most often on the street
• The geometric mean, because house prices grow by multiplying over time

Work out the arithmetic mean

A student sits two papers and scores 60 marks on the first and 80 marks on the second. Work out the arithmetic mean of the two scores.

Work out the weighted mean

The same two papers do not count equally. The first paper has a weight of 2 and the second a weight of 3. Multiply each score by its weight, add the results, and divide by the total of the weights. Work out the weighted mean.

Weighting changes the answer

Look at what just happened. The same two marks gave 70 one way and 72 the other, and neither is wrong. The weighted mean is higher because the better score counted for more. That is the whole idea: a weighted mean says some values matter more, and it is the right tool whenever they genuinely do. One more rule from this specification, and it is worth writing down. When a question says mean with nothing else attached, it means the arithmetic mean. It only means the geometric mean if the question says so.

Work out the geometric mean

A shop multiplies its sales by 2 in one year and by 8 in the next. For two values the geometric mean is the square root of the two multiplied together. Work out the geometric mean of 2 and 8.

Which way does the tail go

Skewness is named for the side the long tail is on, which catches people out because the bulk of the data is on the other side. The order of the three averages follows from it, and knowing that order is usually enough to answer by inspection.

Reading a skewed set

A set of data is positively skewed. Select the TWO statements that must be true.

• The mean is greater than the median
• The long tail of the distribution points to the right
• Most of the values are large ones near the right-hand end
• The mode is greater than the mean

Skewness in a paragraph

A distribution is described as skewed when it is not _____. If the long tail points to the right the skew is _____, and the mean will be greater than the _____. If the long tail points to the left the skew is _____. When a set of data is skewed, the three averages separate, which is why the choice of which one to _____ starts to matter.

What the sign tells you

Higher tier. A question gives you the formula for a skewness coefficient and you work it out as a negative number. What does that tell you?

• The distribution is negatively skewed, so the long tail points to the left
• The distribution is positively skewed, since a negative result reverses the reading
• A mistake has been made, because a skewness coefficient cannot be negative
• The distribution is symmetric, because the negative and positive parts cancel out

Choosing an average, justified

Question: two companies report their staff pay. Company A quotes a mean salary of 42 thousand pounds; company B quotes a median of 31 thousand. Explain why comparing these two figures directly would be misleading, and say what you would ask for. Model answer: the two companies have not quoted the same measure, so the comparison is not like with like. Salary data is usually positively skewed, because a small number of very high earners sit in a long right-hand tail, and the mean is pulled upwards by them while the median is not. Company A quoting a mean will therefore tend to look better than company B quoting a median even if the two workforces are paid similarly. To compare fairly I would ask both companies for the same measure, and I would ask for the median, since it better represents a typical employee when the data is skewed. Notice that the answer names the skew, explains what it does to each measure, and only then makes a recommendation.

Which claim is wrong

Three of these statements about averages are correct. Select the ONE that is not.

• A weighted mean can differ from the arithmetic mean of the same values.
• The median of a set is unchanged if the largest value is made larger still.
• The mode is the only average that can be found for data with no numerical order.
• A geometric mean is the right choice whenever a set contains an extreme value.

Comparing two data sets

You are comparing two sets of data and have to decide what to compare and what to say about it.

• One set is roughly symmetric and the other is strongly positively skewed. What should you compare them on?
• You choose the median for both. What have you gained and what have you given up?
• A colleague asks why you did not use the mean, which uses all the data. What is the honest answer?
• You now want to add one sentence about how spread out each set is. Where does that belong?

Explain your choice

A charity reports the amounts donated to it last month. Most donations were between 5 and 50 pounds, but three were over 10 thousand pounds. Explain which average you would quote to describe a typical donation, and why.

• State which average you would quote
• Describe the shape of this set of data, naming the direction of the skew
• Explain what that skew does to the mean, and why
• Explain why your chosen average is less affected
• Say what would change if the charity wanted to report the total raised rather than a typical donation