Standard Deviation (Leaving Cert Mathematics): Revision Notes

Standard Deviation of Frequency Distribution by Hand

Overview

The standard deviation measures the spread of data around the mean in a frequency distribution. Calculating it by hand for grouped data involves the following steps:

Formula for Standard Deviation

$$s = \sqrt{\frac{\sum f(x - \bar{x})^2}{\sum f}}$$

Where:

• $x$: Midpoint of each class interval.
• $\bar{x}$: Mean of the distribution.
• $f$: Frequency of each class.
• $\sum f(x - \bar{x})^2$: Sum of the squared deviations multiplied by their frequencies.
• $\sum f$: Total frequency.

Steps to Calculate Standard Deviation

1. Calculate the Mean ($\bar{x}$):

• Use the formula:

$$\bar{x} = \frac{\sum (f \times x)}{\sum f}$$

2. Find Deviations:

• Subtract the mean from each midpoint ($x - \bar{x}$).

3. Square the Deviations:

• Square each deviation $((x - \bar{x})^2)$

4. Multiply by Frequency:

• Multiply the squared deviations by their respective frequencies $(f(x - \bar{x})^2)$

5. Sum Up:

• Add all the $f(x - \bar{x})^2$ values to get the total deviation.

6. Calculate Standard Deviation:

• Divide the total deviation by the total frequency $(\sum f)$
• Take the square root.

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Step 3: Compute Deviations and Squares

$x$	$f$	$x • \bar{x}$	$(x • \bar{x})^2$	$f(x • \bar{x})^2$
15	3	−16.5-16.5	272.25	816.75
25	5	−6.5-6.5	42.25	211.25
35	8	3.53.5	12.25	98.00
45	4	13.513.5	182.25	729.00

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Step 4: Total Deviation

$$\sum f(x - \bar{x})^2 = 816.75 + 211.25 + 98.00 + 729.00 = :highlight[1855.00]$$

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Step 5: Standard Deviation

$$s = \sqrt{\frac{\sum f(x - \bar{x})^2}{\sum f}}$$ $$= \sqrt{\frac{1855.00}{20}} = \sqrt{92.75} \approx :highlight[9.63]$$

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Answer: The standard deviation is approximately $:success[9.63]$

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Summary

• The standard deviation measures data spread within a frequency distribution.
• Steps:
  1. Calculate midpoints.
  2. Compute the mean.
  3. Find deviations ($x - \bar{x}$) and square them.
  4. Multiply squared deviations by frequency.
  5. Sum up and divide by total frequency.
  6. Take the square root to get the standard deviation.
• Standard deviation provides valuable insights into data variability and distribution.