Standard Deviation Revision Notes for Leaving Cert Mathematics

Standard Deviation

What is standard deviation?

Standard deviation is one of the most important measures of spread in statistics. It tells us how much variation exists in a data set by measuring how far individual values are from the mean (average).

Think of standard deviation as the average distance that data points are from the mean. A small standard deviation means the data points are clustered close to the mean, while a large standard deviation means the data is spread out over a wider range.

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The Greek letter σ (sigma) is used to represent standard deviation in mathematical notation. This symbol is universally recognised in statistics and mathematics.

The empirical rule

For large populations, standard deviation follows a predictable pattern called the Empirical Rule:

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The Empirical Rule (68-95-99.7 Rule):

68% of values lie within one standard deviation of the mean (between $\bar{x} - \sigma$ and $\bar{x} + \sigma$ )
95% of values lie within two standard deviations of the mean
99.7% (almost all) values lie within three standard deviations of the mean

This rule is fundamental to understanding how data distributes around the mean in normal distributions.

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Real-World Application:

If the average height of Irish men is 177 cm with a standard deviation of 8 cm, then about 68% of men have heights between 169 cm and 185 cm (177 ± 8).

Step-by-step procedure for calculating standard deviation

To find the standard deviation of a set of numbers, follow these 6 steps:

Calculate the mean $(\bar{x})$ of all the numbers
Find each deviation from the mean: $(x - \bar{x})$ for each value
Square each deviation: $(x - \bar{x})^2$
Add up all the squared deviations: $\Sigma(x - \bar{x})^2$
Divide by n (the number of values): $\frac{\Sigma(x - \bar{x})^2}{n}$
Take the square root of the result

The formula is:

$\sigma = \sqrt{\frac{\Sigma(x - \bar{x})^2}{n}}$

Worked example 1: Basic calculation

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Worked Example: Finding Standard Deviation

Find the standard deviation of the numbers: 6, 9, 10, 12, 13

Step 1: Calculate the mean $\bar{x} = \frac{6 + 9 + 10 + 12 + 13}{5} = \frac{50}{5} = 10$

Step 2: Find deviations from mean $(6-10) = -4$ , $(9-10) = -1$ , $(10-10) = 0$ , $(12-10) = 2$ , $(13-10) = 3$

Step 3: Square each deviation
$(-4)^2 = 16$ , $(-1)^2 = 1$ , $(0)^2 = 0$ , $(2)^2 = 4$ , $(3)^2 = 9$

Step 4: Add the squared deviations $16 + 1 + 0 + 4 + 9 = 30$

Step 5: Divide by n $\frac{30}{5} = 6$

Step 6: Take square root $\sigma = \sqrt{6} = 2.45$

Therefore, the standard deviation is 2.45.

Standard deviation from a frequency distribution

When working with frequency distributions, we modify the calculation slightly. Each squared deviation is multiplied by its frequency before adding them up.

The formula becomes:

$\sigma = \sqrt{\frac{\Sigma f(x - \bar{x})^2}{\Sigma f}}$

Where:

$f$ = frequency of each value
$\Sigma f$ = total of all frequencies
$\Sigma f(x - \bar{x})^2$ = sum of frequency × squared deviation

Worked example 2: Frequency distribution

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Worked Example: Standard Deviation from Frequency Distribution

Find the standard deviation for this frequency distribution:

Variable (x)	1	2	3	4	5	6
Frequency (f)	9	9	6	4	7	3

Step 1: Find the mean $\bar{x} = \frac{9×1 + 9×2 + 6×3 + 4×4 + 7×5 + 3×6}{9+9+6+4+7+3}$

$\bar{x} = \frac{9 + 18 + 18 + 16 + 35 + 18}{38} = \frac{114}{38} = 3$

Step 2: Create a calculation table

Table

Step 3: Apply the formula $\sigma = \sqrt{\frac{104}{38}} = \sqrt{2.74} = 1.65$

Therefore, the standard deviation is 1.65.

Using a calculator for standard deviation

Scientific calculators can significantly reduce calculation time. Here's how to use a Casio fx-83ES calculator:

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For simple data sets:

Press MODE then 2 (for statistics mode)
Select 1 for 1-VAR
Input your numbers using the = key after each one
Press AC, SHIFT, 1, 5, 2, = for the mean
Press AC, SHIFT, 1, 5, 3, = for standard deviation

For frequency distributions:

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Calculator Steps for Frequency Data:

Press MODE then 2 (statistics mode)
Select 1 for 1-VAR
Input each variable and its frequency
Use the navigation keys to move between X and FREQ columns
Press the same key sequences as above to get results

Worked example 3: Calculator method

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Worked Example: Using Calculator Method

Using the calculator for the data set 5, 3, 1, 8, 2:

Key sequence:

MODE → 2 → 1 → 5 = 3 = 1 = 8 = 2 =

AC → SHIFT → 1 → 5 → 2 → = (gives mean = 3.8)

AC → SHIFT → 1 → 5 → 3 → = (gives σ = 2.5)

Worked example 4: Golf birdies

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Worked Example: Real-World Application

A golfer's birdie count per round over 25 rounds:

No. of birdies	0	1	2	3	4	5	6
Frequency	5	6	4	6	3	1	0

Using the calculator method:

Mean = 2.0 birdies per round
Standard deviation = 1.5

This tells us that most rounds had between 0.5 and 3.5 birdies (within one standard deviation of the mean).

Exam tips

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Essential Exam Strategies:

Always show your working - even when using a calculator, write down the key steps
Round appropriately - usually to 1 or 2 decimal places as specified
Check your answer makes sense - standard deviation cannot be negative
For grouped data, use the mid-interval values as your x-values
Remember the empirical rule - it's often tested in exam questions

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Key Points to Remember:

Standard deviation measures spread - how scattered data is around the mean
Small σ = data clustered together, large σ = data spread out
The formula is $\sigma = \sqrt{\frac{\Sigma(x - \bar{x})^2}{n}}$ for simple data
For frequency distributions, multiply by frequencies: $\sigma = \sqrt{\frac{\Sigma f(x - \bar{x})^2}{\Sigma f}}$
Use your calculator efficiently - learn the key sequences to save time in exams

Standard Deviation (Leaving Cert Mathematics): Revision Notes