Standard Deviation
Standard deviation is a measure of statistical dispersion that quantifies the variability or spread of a set of data around its mean. It is calculated by the square root of the variance.
Formula:Standard deviation (ฯ) = โ[Variance (Var) = ฮฃ(x - ฮผ)ยฒ / n - 1]
where:* ฯ is the standard deviation* Var is the variance* x is each element in the data set* ฮผ is the mean of the data set* n is the number of elements in the data set
Interpretation:
- Standard deviation measures the degree of variation in a data set.
- A high standard deviation indicates a wide range of values, while a low standard deviation indicates a narrow range of values.
- The standard deviation is an important measure of data dispersion, alongside the range, variance, and coefficient of variation.
Significance:
- Standard deviation is used to:
- Describe the variability of a data set.
- Compare the variability of different data sets.
- Calculate confidence intervals and hypothesis tests.
- Evaluate the performance of statistical models.
Example:
“`Data: [10, 12, 14, 16, 18]
Mean (ฮผ) = (10 + 12 + 14 + 16 + 18) / 5 = 16
Variance (Var) = [(10 – 16)ยฒ + (12 – 16)ยฒ + (14 – 16)ยฒ + (16 – 16)ยฒ + (18 – 16)] / (5 – 1) = 8
Standard Deviation (ฯ) = โ8 = 2.828“`
Therefore, the standard deviation of the data set is 2.828.
FAQs
What is standard deviation?
Standard deviation is a measure of the spread or dispersion of a set of numbers. It shows how much the numbers in a data set deviate from the mean (average). A small standard deviation means the numbers are close to the mean, while a large one indicates that they are spread out over a wide range of values.
What is an example of standard deviation in real life?
In real life, standard deviation can be used to measure variability in test scores. For example, in a classroom, if most students score close to the average, the standard deviation will be small. If the scores vary widely, with some students scoring very high and others very low, the standard deviation will be large.
