The mean and the spread: comparing investments

In investments, the standard deviation of the returns is a measure of risk: the larger it is, the further a given year’s return can be from the mean – a loss too.

The returns of two shares over five years, in percent:

• Share A: the mean is 7%, the standard deviation $\displaystyle \sqrt{2}\approx1.41$ percent.
• Share B: the mean is 8%, the standard deviation $\displaystyle \sqrt{86}\approx9.27$ percent.

Share B has the higher mean return, but in one year it lost 5%, and in another it earned nothing.

A high mean return usually comes with a high risk, so the choice depends on who invests and how much time they have:

• Someone saving for 20 years can wait for the bad years to pass – share B, with the higher mean, suits them better.
• Someone who needs the money next year, for driving lessons or for tuition fees, cannot afford a year of loss – share A, with the small standard deviation, suits them better.

Two groups with the same mean can be very different: the standard deviation shows how different.

The monthly pay of five workers in each company, in thousands of shekels:
Company A: $\displaystyle 9,\ 10,\ 10,\ 11,\ 10$
Company B: $\displaystyle 6,\ 8,\ 10,\ 12,\ 14$

• In both companies the mean is 10 thousand ₪.
• In company A the standard deviation is $\displaystyle \sqrt{0.4}\approx0.63$ thousand ₪: everyone earns about the same.
• In company B the standard deviation is $\displaystyle \sqrt{8}\approx2.83$ thousand ₪: there are big differences between the workers.

When the same number is added to all the data, the mean grows by that number, and the standard deviation does not change: all the data move together, and their distances from the mean stay the same.

All of company B’s workers got a rise of 500 ₪ (half a thousand):
The new pay: $\displaystyle 6.5,\ 8.5,\ 10.5,\ 12.5,\ 14.5$

• The mean: 10.5 thousand ₪ – up by 0.5.
• The deviations from the mean: $\displaystyle -4,\ -2,\ 0,\ 2,\ 4$ – as before, so the standard deviation stayed $\displaystyle \approx2.83$.

When all the data grow by the same percent, both the mean and the standard deviation grow by that percent: every deviation from the mean is multiplied by the same factor.

All of company B’s workers got a 10% rise – each pay was multiplied by 1.1:
The new pay: $\displaystyle 6.6,\ 8.8,\ 11,\ 13.2,\ 15.4$

• The mean: $\displaystyle {10\cdot1.1}={11}$ thousand ₪.
• The standard deviation: $\displaystyle \sqrt{8}\cdot1.1\approx3.11$ thousand ₪ – the gaps between the workers grew.

The returns of two savings funds over five years, in percent:
Fund A: $\displaystyle 3,\ 4,\ 4,\ 5,\ 4$
Fund B: $\displaystyle 8,\ -4,\ 12,\ 2,\ 7$

Work out each fund’s mean return.

Fund A:

Fund B:

Work out each fund’s standard deviation. Round to two decimal places.

Fund A:

Fund B:

Which fund suits a saver who needs the money in a year, and which a saver who needs it in 15 years? Explain.

The monthly rent of five flats in two neighbourhoods, in shekels:
Neighbourhood A: $\displaystyle 3200,\ 3500,\ 3400,\ 3600,\ 3300$ – the mean is 3,400 ₪.
Neighbourhood B: $\displaystyle 2500,\ 4500,\ 3000,\ 4000,\ 3500$ – the mean is 3,500 ₪.

Work out the standard deviation in neighbourhood A. Round to two decimal places.

Work out the standard deviation in neighbourhood B. Round to two decimal places.

A student is looking for a flat on a budget of exactly 3,500 ₪. In which neighbourhood is it easier to guess in advance what a flat will cost? Explain.

In a clothes shop, the workers’ mean pay is 8,000 ₪, and the standard deviation 1,200 ₪.
The management offers two choices of rise:
Choice A – 400 ₪ for every worker.
Choice B – 5% for every worker.

What will the standard deviation be after choice A?

What will the mean be after choice B?

What will the standard deviation be after choice B?

Which choice will the workers who earn less than 8,000 ₪ pick? Explain.

An investment fund’s advert says: “A mean return of 9% a year over the last five years!”

What other figure should you ask for before investing in the fund? Explain why.