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Note. When we square negative numbers, as well as positive numbers, we obtain positive answers. For example, (—4)2 = 16 and (+4)2 = 16. So if we are solving equations by taking the square root, we must write down both the positive and the
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negative solutions. The solutions of the equation x2 = 16 are x = +4 and x = —4. These two answers are usually expressed as x = ±4.
References: Indices, Inverse Operations, Pythagoras’ Theorem, Squaring.
This is an operation that multiplies any number, or term, by itself. For example, when we square 7, the result is 7x7 =49. This can also be written as 72, which we say in words as seven squared.
Example 1. Square 2x + 3.
Solution. The square of 2x + 3 is (2x + 3)2. Write (2x + 3)2 = (2x + 3) x (2x + 3)
= 4x2 + I2x + 9. Expanding the brackets
Example 2. Find the square 5xy3.
Solution. The square of 5xy3 is (5xy3)2. Write
(5xy3)2 = 5xy3 x 5xy3 a2 = a x a
= 25x2y6 Using laws of algebra
References: Algebra, Expanding Brackets, Square Numbers.
Under this entry we study spread, range, and standard deviation, which are three statistical terms. Range and standard deviation both measure the spread of a set of data. Another measure of spread uses quartiles. The terms spread, range, and standard deviation are illustrated in the example that follows.
Luke plays junior rugby for Plato High School, and Will plays for Socrates High School. The weights of some players are very important in the game of rugby, and the weights of each player in the two teams are shown in the table in kilograms. There are 15 players in each team. The total weight of each team and the mean weight of each team are calculated as shown.
Plato rugby team
Weights, in kg, of individual players: 56, 59, 62, 63, 66, 69, 70, 75, 78, 80, 83, 84, 85, 87, 93 Total weight = 1110 kg, mean weight = 1110 = 15 = 74 kg Socrates rugby team
Weights, in kg, of individual players: 62, 62, 63, 64, 69, 70, 72, 76, 77, 77, 79, 81, 85, 86, 87 Total weight = 1110 kg, mean weight = 74 kg
What a coincidence! The mean weight of each team is the same.
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We now compare the “spread” of the weights of each team, but first give an introduction to the term spread. When you are spreading jam on your toast you can spread it evenly over the whole slice or you can “load it up” in the middle and save the most jam for the final mouthful. Similarly, the weights of the players in a rugby team can be well spread out or they can be “clustered” closely together. We could say that the “center” of the range of weights is the mean. The weights of the players from each team are plotted on a number line in order to compare how spread out the data are (see the figure).
Mean weight, kg ♦
PLATO O O OO o OO o o o ooo o O
5 60 65 70 75 80 85 90 95
SOCRATES o o o O OO o o o o o o OOO
5 60 65 70 75 80 85 90 95
It can easily be seen that the weights of the Plato players are more spread about their mean than the weights of the Socrates players. One of the ways of measuring spread is called the range. The range is the difference between the largest value and the smallest value:
Plato: Range = 93 — 56
Socrates: Range =87 — 62
We therefore conclude that the distribution of weights of the Plato players is more spread out than that for the Socrates players, since they have a greater range. Since range only involves 2 players and ignores the other 13 players, it does not give us an accurate picture of how spread out the whole data are. A more reliable figure to indicate spread is called standard deviation. This figure tells us how spread out the weights are from the mean by including every person’s weight in the calculation. The steps involved in calculating standard deviation are as follows:
1. Calculate the mean weight of the players, which is x = 74 in this example.
2. Calculate the deviation (difference) of each weight (x) from the mean x, which is (x — x).
3. Square the deviation of each weight from the mean, which is (x — x)2 for each player.
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4. Add together all these terms (x — x)2, which is written as EC* — x)2. Here E is the capital letter of the Greek alphabet, and means “add together.”
5. Divide this total by the number of players in the team, which is
EC* - *)2
6. Take the square root of this answer to give the standard deviation of the weights of the players on the team. The standard deviation can be described as the square root of the mean of the squares of the deviations. The symbol for standard deviation is cr, which is lower case Greek sigma:
These steps for calculating the standard deviation can be clearly set out in a table as shown, for each team of rugby players. Alternatively, the standard deviation can be found using a scientific calculator, with the processes explained in the calculator’s handbook. In the table, the first line in each column sets out the steps of working for that column.