Welcome to Sections 4.2 & 4.3 Part 2!
Hello Math Students! In this section, we're continuing our journey into understanding data. We'll be focusing on measures of variation, including standard deviation, coefficient of variation and z-scores. Let's dive in!
Understanding Standard Deviation
Standard deviation ($s$ or $\sigma$) tells us how spread out the data is around the mean. A smaller standard deviation indicates that data points are clustered closely around the mean, while a larger standard deviation indicates a wider spread.
Here's an example. Consider the following data set:
81, 81, 82, 83, 85, 86, 91, 93, 96, 97, 99, 99
The mean is 89.4, and the standard deviation is 6.9.
Now, let's add 20 to each data value:
101, 101, 102, 103, 105, 106, 111, 113, 116, 117, 119, 119
The new mean is 109.4. Interestingly, the standard deviation remains the same at 6.9! Adding a constant to each data point shifts the entire distribution but doesn't change the spread.
Coefficient of Variation
The Coefficient of Variation (CV) is a relative measure of variation. It's useful for comparing the variability of datasets with different units or different means. The formula for CV is:
- For population data: $$CV = \left(\frac{\sigma}{\mu} \cdot 100\right)\%$$
- For sample data: $$CV = \left(\frac{s}{\bar{x}} \cdot 100\right)\%$$
Where:
- $\sigma$ is the population standard deviation
- $\mu$ is the population mean
- $s$ is the sample standard deviation
- $\bar{x}$ is the sample mean
Example: Vitamin C Content
Let's compare two brands of vitamin C:
| Brand A (500 mg) | Brand B (250 mg) | |
|---|---|---|
| Mean ($\bar{x}$) | 500 mg | 250 mg |
| Standard Deviation ($s$) | 10 mg | 7 mg |
Calculating the CV:
- Brand A: $CV = (10 / 500) * 100 = 2\%$
- Brand B: $CV = (7 / 250) * 100 = 2.8\%$
Brand A has a lower CV, indicating that it produces tablets more consistently as advertised.
Z-Scores
A z-score transforms a data value into the number of standard deviations that value is from the mean. It allows us to compare values from different datasets on a standardized scale. The formula is:
$$z = \frac{x - \mu}{\sigma}$$
Where:
- $x$ is the data value
- $\mu$ is the mean
- $\sigma$ is the standard deviation
Example:
If the mean daily sales of a diner is $4500 with a standard deviation of $750:
- 68% of the time, the sales will be in the range of $4500 ± $750
- 95% of the time, the sales will be in the range of $4500 ± 2*$750
- 99.7% of the time, the sales will be in the range of $4500 ± 3*$750
Remember: When answering the questions above, we assumed the daily sales were normally distributed.
Keep practicing, and you'll master these concepts in no time! Good luck!