## Measures of Dispersion in Statistics - StDev

Another Example

Assume that today’s date is July 1.

John has $1,000. He earmarks this sum for a debt of $1,000 due for repayment on August 1, i.e., in one month. During July, he decides to invest the money in shares.

He can buy shares of either Bank A or Bank B with the money.

In order to decide which shares to buy, John examined the performance of these shares over the prior six months (between January and June). The performance of the shares is measured in terms of monthly profit expressed in percentages.

The Monthly Profits From each of the Banks in Terms of Percentages:

Month | Shares of Bank A | Shares of Bank B |

January | 1% | 3% |

February | 1% | -2% |

March | 1% | 5% |

April | 1% | -3% |

May | 1% | 8% |

June | 1% | -5% |

Average profit per month | 1% | 1% |

According to these figures, the average profit earned on each of the shares is 1% per month (i.e., $10 per month, if the investment is $1,000). When we assess the dispersal, however, we can see that the profit on the share of Bank A is very stable, while the profit on the share of Bank B is unstable and cannot be counted on.

In the theory of finance, a share with a wide dispersal of profit is called “a highly volatile share”, and is considered a riskier investment particularly if the investment is short-term, as in John’s case.

When the two shares yield the same average monthly profit, we will prefer to invest in less volatile shares - the shares with more stable monthly profits.

In this case, we do not need to calculate the StDev in order to determine the measures of dispersion in statistics because it is obvious that Bank B’s share is more dispersed (more volatile.)

Bank A’s share has no dispersal, the StDev is therefore zero.

If, however, the monthly profit figures are as follows:

Month | Shares of Bank A | Shares of Bank B |

January | 5% | 3% |

February | 1% | -2% |

March | -6% | 5% |

April | 10% | -3% |

May | -7% | 8% |

June | -2% | -5% |

Average profit per month | 1% | 1% |

In this case, the volatility of both shares is high, and there is need for a “volatility meter” in order to decide which share is more stable. Since volatility in essence arises from dispersal, we will measure the dispersal. The tool we have for measuring dispersal is the standard deviation.

We will calculate the StDev of Bank A’s share:

The Value | The Average | The Difference Between the Value and the Average | The Square of the Difference |

(1) | (2) | (3) = (1) – (2) | (4) = (3)2 |

5 | 1 | 4 | 16 |

1 | 1 | 0 | 0 |

-6 | 1 | -7 | 49 |

10 | 1 | 9 | 81 |

-7 | 1 | -8 | 64 |

-2 | 1 | -3 | 9 |

We will calculate the average of the squares of the differences (column 4):

(16+0+49+81+64+9) / (6) = 36.5

The StDev reflects the square root of 36.5 = 6.041.

We will calculate the StDev of Bank B’s share:

The Value | The Average | The Difference Between the Value and the Average | The Square of the Difference |

(1) | (2) | (3) = (1) – (2) | (4) = (3)2 |

3 | 1 | 2 | 4 |

-2 | 1 | -3 | 9 |

5 | 1 | 4 | 16 |

-3 | 1 | -4 | 16 |

8 | 1 | 7 | 49 |

-5 | 1 | -6 | 36 |

We will calculate the average of the squares of the differences (Column 4):

(4+9+16+16+49+360 / (6) = 21.67

The StDev is the square root of 21.67 = 4.655.

The standard deviation of Bank B’s share is smaller, which means that the degree of dispersal of its profit is smaller - less volatile, less risky

John will choose to invest in Bank B’s share.