# STANDARD DEVIATION AS A STATISTICAL CONCEPT

The term ‘standard deviation’ was first used in writing by Karl Pearson in 1894(Dodge, Yadolah 2003) following his use of it in lectures. This was as a replacement for earlier alternative names for the same idea like ‘mean error’ (Miller, Jeff- Wikipedia). Oxford Dictionary of Banking and Finance (2008) defines standard deviation as a measure of the dispersion of statistical data. It measures how far apart data are from the average of the data. In a probability distribution, standard deviation can be used to estimate the tendency of data to be spread out. It shows how much variation there is from the ‘average’ (also known as mean). A small standard deviation indicates that the data points tend to be clustered around the mean whereas a large standard deviation indicates that the data are far from the mean.

From the above explanation, we see that standard deviation derives its relevance from the ‘average’ (subsequently referred to as mean). An average represents a typical value. For example the average height of the children in a family, the average temperature in a particular day, average price of commodities in the market. The average is usually calculated based on past records. The average is calculated by adding up the values in a sample population divided by the number of data examined. For the sake of analysis, let us assume we want to find the average number of customers that visit the local branch of a bank in day. We have the following data.

Days

Monday

Tuesday

Wednesday

Thursday

Friday

Number of customers

1200

800

580

620

1000

The simple average of the above data is

1200+800+580+620+1000 ÷ 5 = 840

This means about 840 customers visit the bank’s branch per day.

ESTIMATING STANDARD DEVIATION

Statistically, Standard deviation is the square root of the variance of a data set (Elton et al, 2007). Standard deviation is calculated by first calculating the mean of a data set. Each of the data is then subtracted from the mean to get the individual data variation from the mean value. The variations are squared and added together. The sum of the squared variations is the data variance. Then, we find the square root of the variance to arrive at the standard deviation of the data set. It is mathematically expressed as:

(e-zine on Wikipedia)

Where s = standard deviation, ∑= summation, (x-x)² = variation from the mean, n = number of data in the set.

Using our example in the previous paragraph, the average (mean represented by x bar) is 840. We calculate standard deviation thus:

Day

No of customers

x

Deviation from the mean

( x-x bar)

Square of the deviation

(x-x bar)²

Monday

1200

360

129600

Tuesday

800

-40

1600

Wednesday

580

-260

67600

Thursday

620

-220

48400

Friday

1000

160

25600

Total

272800

S= √272800

5-1

= 261.15

USE OF STANDARD DEVIATION

Standard deviation is used for measuring variability in many endeavours of life. Statisticians used it to measure the level of confidence in statistical conclusions (). For example in population census, there is an expected margin of error in the outcome of the exercise. If the outcome is within the acceptable standard deviation, the result can be adopted. If it is outside the acceptable standard deviation, the result is rejected. In science, researchers commonly report the standard deviation of experimental data (). Effects that fall far outside the range of standard deviation are considered ‘statistically significant’ ().

Our emphasis in this text is finance. Standard deviation is a representation of the risk associated with a given security () such as stocks, bonds, and properties. In modelling stock price performances for instance, standard deviation ‘describes how prices are dispersed around an average value’. Using two standard deviations ensures that 95% of the price data will fall between the two trading bands (Murphy, J. 1999).

Risk is a key factor in making investment decisions. It is used to determine the variations on return on assets (ROA). Basically, two risks are involved in investing money; the risk of default by the borrowing company, and the risk of reduced purchasing power of the future expected return due to economic inflation. An investor has to factor in this uncertainty in the expected returns. This is what standard deviation measures. Depending on the risk appetite of an investor, the traditional model is that as risk increases, the expected returns on the asset increases.

For example, let us assume that Mr Bright is considering investing in either of the stocks of Barclays bank and British Airways. The prices of the stocks are £283.05 and £283.40 respectively (prices as quoted on London Stock Exchange as at 22-10-2010). Barclays has an average return of 15% over the last decade with a standard deviation of 20 percentage points and British Airways, over the same period, has an average return of 17% with a standard deviation of 30 percentage points. On the basis of risk and return, Mr Bright may decide that Barclays is better because British Airways extra 2% points of return is not worth the additional 10 percentage point standard deviation (greater risk). British Airways is likely to fall short of the initial investment more often than Barclays under the same circumstances, and it is estimated to return only 2% more on average. Barclays is expected to earn about 10%, plus or minus 20pp- about 67% of future year returns. Rationally, in considering more extreme possible future returns, Mr Bright expects results of up to 10% plus or minus 60pp, or a range from 70% to (-)50%, which includes outcome for three standard deviations from the average return- that is about 99.7% of probable returns.

ESTIMATING THE RISK OF INVESTMENT

An investment is the current commitment of money or other resources in the expectation of reaping future benefits (Bodie et al, 2009). The basis of investment is expected returns. An investor commits his resources with an eye on what it will yield in future date. Therefore, the key issue in investment is estimating expected returns (Fischer Black, 1993). However, there is a price to pay for the expected returns- risk. Bodie et al, (2009) argues further that ‘the risk-return trade-off and the efficient pricing of financial assets are central to the investment process’.

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