Understanding Standard Deviation

The standard deviation is a measure of the amount of variation or dispersion of a set of data points. A low standard deviation indicates that the data points tend to be close to the mean of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values. In this problem, we are asked to find the standard deviation of the data set: 5, 3, 4, 7.

There are slightly different formulas for population standard deviation ($\sigma$) and sample standard deviation ($s$). Based on the options provided and common practice when a small data set is given without context, the calculation often implies the population standard deviation formula or the formula $\sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}$. We will proceed with this interpretation.

Steps to Calculate Standard Deviation

To find the standard deviation of the given data, we follow these steps:

1. Calculate the mean ($\mu$) of the data.
2. Find the deviation of each data point from the mean ($x_i - \mu$).
3. Square each deviation ($(x_i - \mu)^2$).
4. Sum the squared deviations ($\sum (x_i - \mu)^2$).
5. Calculate the variance ($\sigma^2$) by dividing the sum of squared deviations by the number of data points ($n$).
6. Calculate the standard deviation ($\sigma$) by taking the square root of the variance.
7. Round the result to two decimal places as required.

Step 1: Calculate the Mean

The mean ($\mu$) is the sum of all data points divided by the number of data points ($n$).

Data points ($x_i$): 5, 3, 4, 7

Number of data points ($n$): 4

Sum of data points ($\sum x_i$): $5 + 3 + 4 + 7 = 19$

Mean ($\mu$):

$\mu = \frac{\sum x_i}{n} = \frac{19}{4} = 4.75$

The mean of the data is 4.75.

Step 2: Calculate Deviations from the Mean

Subtract the mean (4.75) from each data point ($x_i - \mu$).

• $5 - 4.75 = 0.25$
• $3 - 4.75 = -1.75$
• $4 - 4.75 = -0.75$
• $7 - 4.75 = 2.25$

Step 3: Square the Deviations

Square each of the deviations calculated in the previous step ($(x_i - \mu)^2$).

• $(0.25)^2 = 0.0625$
• $(-1.75)^2 = 3.0625$
• $(-0.75)^2 = 0.5625$
• $(2.25)^2 = 5.0625$

Step 4: Sum the Squared Deviations

Add up all the squared deviations ($\sum (x_i - \mu)^2$).

Sum of squared deviations = $0.0625 + 3.0625 + 0.5625 + 5.0625 = 8.75$

Step 5: Calculate the Variance

The variance ($\sigma^2$) is the average of the squared deviations. Using the population variance formula:

$\sigma^2 = \frac{\sum (x_i - \mu)^2}{n}$

$\sigma^2 = \frac{8.75}{4} = 2.1875$

The variance is 2.1875.

Step 6: Calculate the Standard Deviation

The standard deviation ($\sigma$) is the square root of the variance.

$\sigma = \sqrt{\sigma^2} = \sqrt{2.1875}$

Calculating the square root:

$\sigma \approx 1.478935$

Step 7: Round the Result

We need to round the standard deviation to two decimal places.

$\sigma \approx 1.478935$ rounded to two decimal places is 1.48.

Final Answer Summary

The standard deviation of the data set 5, 3, 4, 7 is approximately 1.48 when rounded to two decimal places.

Variance ($\sigma^2$) = $\frac{8.75}{4} = 2.1875$

Standard Deviation ($\sigma$) = $\sqrt{2.1875} \approx 1.48$

Revision Table - Key Statistics Concepts

Additional Information - Standard Deviation Use

The standard deviation is a fundamental statistic used in many fields. Here are some key points about its use:

• Data Comparison: It allows comparing the spread of different data sets, even if they have similar means.
• Quality Control: Used in manufacturing to monitor the consistency of products. A smaller standard deviation indicates less variation and higher quality consistency.
• Finance: Used to measure the volatility or risk of investments. A higher standard deviation means greater price swings and higher risk.
• Statistical Inference: It is a key component in many statistical tests and confidence interval calculations.
• Normal Distribution: For data that follows a normal distribution, about 68% of the data falls within one standard deviation of the mean, about 95% within two standard deviations, and about 99.7% within three standard deviations. This is known as the empirical rule or the 68-95-99.7 rule.