Statistics

Formula for Standard Deviation

We will learn how to calculate the standard deviation by using the two previous examples:

Sample A: 9, 8, 7, 6, 5, 4, 3, 2, 1

Sample B: 9, 5, 5, 5, 5, 5, 5, 5, 1

There is no point in compiling a frequency table for Sample A since each value appears exactly once.

We will therefore calculate the standard deviation from the sample given in the list.

First, we need to calculate the average: (1+2+3+4+5+6+7+8+9) / (9) = 5

Now we subtract the average from each of the values, and then calculate the square of the result:


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

1

5

-4

16

2

5

-3

9

3

5

-2

4

4

5

-1

1

5

5

0

0

6

5

1

1

7

5

2

4

8

5

3

9

9

5

4

16

We can now calculate the average of the numbers in column 4 (the average of the square of the differences): (16+9+4+1+0+1+4+9+16) / (9) = 6.67

The standard deviation represents the square root of the number that we received.

In our case:

σ = The square root of 6.67 = 2.582

For Sample B, we will first compile a frequency table in order to calculate the average:


The Value

The Frequency

The Relative Frequency

The Contribution to the Average

1

1

11.11%

0.1111

5

7

77.78%

3.8889

9

1

11.11%

1

9

100%

5

The average is 5.

We will now compile another frequency table to calculate the standard deviation:

The Value

The Frequency

The Relative Frequency

The Average

The Difference Between the Value and the Average

The Square of the Difference

The Contribution to the Dispersal

(1)

(2)

(3)

(4)

(5) = (1) – (4)

(6) = (5)2

(7) = (3) x (6)

1

1

11.11%

5

1 – 5 = -4

(-4)2 = 16

11.11% x 16 = 1.78

5

7

77.78%

5

5- 5 = 0

02 = 0

77.78% x 0 = 0

9

1

11.11%

5

9 – 5 = 4

42 = 16

11.11% x 16 = 1.78

9

100%

3.56


The standard deviation: = The square root of 3.56 = 1.886.

As we expected, this is below the standard deviation of Sample A. We have seen that this calculation has also verified our intuition that Sample A is more widely dispersed than Sample B.


The Standard Deviation


 The Significance of the Standard Deviation as a Measure of Dispersal in the Decision-making Process

StDev is the shorthand for Standard Deviation

Example:

In country A, a competition is held every year to choose the champion half-court basketball shooter. Every basketball team sends one representative to the competition.

Each contestant in the competition shoots 10 times from mid-court.

The person who will makes the most baskets receives a prize of $1 million, and his coach gets the same amount.

Spurs basketball team, the coach chooses his representative from among four outstanding players by using the following method: He has each player take five rounds of 10 half-court shots each. He chooses the player with the highest average in the five rounds.

The following table shows the results of the rounds of shots by the players:

Player 1 Player 2 Player 3 Player 4
Results of round 1 1 basket 3 baskets 6 baskets 8 baskets
Results of round 2 0 baskets 2 baskets 4 baskets 2 baskets
Results of round 3 2 baskets 4 baskets 6 baskets 8 baskets
Results of round 4 0 baskets 0 baskets 4 baskets 1 basket
Results of round 5 2 baskets 6 baskets 5 baskets 6 baskets
Total baskets 5 baskets 15 baskets 25 baskets 25 baskets
Average per round 1 basket 3 baskets 5 baskets 5 baskets


The coach chooses the player with the highest average of making baskets, but there are two such players: Player 3 and player 4.

The coach of the Spurs must choose between player 3 and player 4, and asks you to help him make a choice.

He also tells you two important facts:

  1. The winner of last year’s competition made an average of 4 baskets per round.

  2. The coach will win $1 million if his representative wins.

Even without calculating the StDev of the shots of the two players, it is easy to see that the dispersal of player 3 is smaller than the dispersal of player 4. The significance of the differences in the measures of dispersion in statistics is that the stability of player 3 is greater, and he will therefore put the coach’s prize in less jeopardy. Had player no. 3 demonstrated his current skill in last year’s competition; he had a good chance of winning, or at least sharing first place, since in no round did he score less than four baskets.

If, however, it is known that a player who always scores on 7 out of 10 shots will participate in the competition, then we will prefer to send player 4 to the competition.

If he has a hot hand on the day of the competition, then he will win; if he does not, he will lose. Player 3, however, in spite of his stability will never make more than six shots in any given round, and so he is therefore sure to lose.

Formula for Standard Deviation 533Formula for Standard Deviation