Standard Deviation

Standard

is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. It represents how spread out the values in a dataset are from the mean (average) of the dataset.

Interpretation

1. Small Standard Deviation:

   • Indicates that the data points are close to the mean.
   • Implies less variability and more consistency within the dataset.
   • Example: In a class where most students score similarly on a test, the standard deviation of the scores will be small.

2. Large Standard Deviation:

   • Indicates that the data points are spread out over a wider range of values.
   • Implies greater variability and less consistency within the dataset.
   • Example: In a class where students’ test scores vary significantly, the standard deviation will be large.

Formulas

Population Standard Deviation

$$\sigma =\sqrt{\frac{{\sum }_{i=1}^N(x_i-\mu {)}^2}{N}}$$

Where:

• $\sigma$ is the population standard deviation.
• $x_i$ represents each data point in the population.
• $\mu$ is the population mean.
• $N$ is the number of data points in the population.

Sample Standard Deviation

$$s=\sqrt{\frac{{\sum }_{i=1}^n(x_i-\bar{x}{)}^2}{n-1}}$$

Where:

• $s$ is the sample standard deviation.
• $x_i$ represents each data point in the sample.
• $\bar{x}$ is the sample mean.
• $n$ is the number of data points in the sample.

Applications

1. Data Analysis:

   • Used to measure the volatility or variability of data.
   • A smaller standard deviation indicates that data points are close to the mean, while a larger standard deviation indicates that data points are spread out.

2. Quality Control:

   • Used to monitor the consistency of product quality.
   • A smaller standard deviation indicates consistent product quality, while a larger standard deviation indicates more variation in quality.

3. Risk Assessment:

   • In finance, used to measure the volatility of investment returns.
   • A higher standard deviation indicates higher risk due to greater variability in returns, while a lower standard deviation indicates lower risk and more stable returns.

Example Calculations

Population Standard Deviation Example

Consider a population with data points: 2, 4, 4, 4, 5, 5, 7, 9.

1. Calculate the mean ($\mu$):

   $$\mu =\frac{2+4+4+4+5+5+7+9}{8}=5$$

2. Calculate each deviation from the mean, square it, and sum:

   $$\sum (x_i-\mu {)}^2=(2-5{)}^2+(4-5{)}^2+(4-5{)}^2+(4-5{)}^2+(5-5{)}^2+(5-5{)}^2+(7-5{)}^2+(9-5{)}^2=4+1+1+1+0+0+4+16=27$$

3. Divide by the number of data points ($N$) and take the square root:

   $$\sigma =\sqrt{\frac{27}{8}}=\sqrt{3.375}\approx 1.84$$

Sample Standard Deviation Example

Consider a sample with data points: 2, 4, 4, 4, 5, 5, 7, 9.

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

   $$\bar{x}=\frac{2+4+4+4+5+5+7+9}{8}=5$$

2. Calculate each deviation from the mean, square it, and sum:

   $$\sum (x_i-\bar{x}{)}^2=(2-5{)}^2+(4-5{)}^2+(4-5{)}^2+(4-5{)}^2+(5-5{)}^2+(5-5{)}^2+(7-5{)}^2+(9-5{)}^2=4+1+1+1+0+0+4+16=27$$

3. Divide by the number of data points minus one ($n-1$) and take the square root:

   $$s=\sqrt{\frac{27}{7}}\approx 1.96$$