Standard Deviation Calculator

This standard deviation calculator computes the standard deviation of a set of numbers and provides additional information, including the mean and variance. The calculator also determines the confidence interval of the dataset for various confidence levels and provides a frequency distribution table.

To use this calculator, enter numbers separated by commas. Select whether the numbers represent a population or a sample, and click "Calculate."

The Standard Deviation

The standard deviation is a statistical measure that defines the degree of spread or variability of a given data set. It provides the aggregated average distance of the data points from the mean. A smaller standard deviation indicates that data points are clustered close to the mean, while a large standard deviation suggests they are more dispersed. The standard deviation is the square root of another measure of spread called the variance.

The calculation depends on whether the dataset represents an entire population or a sample. If it represents all data points of interest, it is the population standard deviation. If the dataset is a sample from a larger population, it is the sample standard deviation.

The Population Standard Deviation

The population standard deviation (σ) is calculated when the dataset includes all observations under consideration. The formula is:

σ = √[ Σ(xᵢ - μ)² / N ]

• Σ: A Greek letter (Sigma) denoting summation.
• xᵢ: Each individual data point.
• μ: The population mean.
• N: The population size.

Example of Population Standard Deviation Calculation

An investment manager analyzes the volatility of a stock's daily closing prices for the previous month. The full dataset is available, so it's a population. The manager considers a stock "too risky" if its standard deviation is greater than or equal to its mean. Stock prices (in USD): 1.31, 1.30, 1.36, 1.40, 1.40, 1.41, 1.27, 1.19, 1.15, 1.12, 0.99, 1.00, 0.97, 0.94, 0.88, 0.90, 0.86, 0.88, 0.80, 0.81.

1. First, calculate the mean (μ): 1.097.
2. Next, calculate the variance (σ²), which is the average of the squared differences from the mean: 0.045031.
3. Finally, take the square root of the variance to get the standard deviation: σ ≈ 0.21.

Since the standard deviation (0.21) is less than the mean (1.097), the manager will not consider this stock "too risky."

The Sample Standard Deviation

The sample standard deviation (s) is used when the dataset is a smaller subset of a larger population. The formula uses n-1 in the denominator, which provides a better estimate of the population's standard deviation.

s = √[ Σ(xᵢ - x̄)² / (n - 1) ]

• x̄: The sample mean.
• n: The sample size.

Example of Sample Standard Deviation Calculation

Now, assume the manager only has closing prices for 5 random days: 1.31, 1.40, 0.86, 0.88, 1.40. The calculation is now for a sample.

1. First, calculate the sample mean (x̄): 1.17.
2. Next, calculate the sample variance (s²): 0.0764.
3. Finally, take the square root of the variance: s ≈ 0.28.

Margin of Error

The standard deviation is crucial for calculating the margin of error, which defines the width of a confidence interval. The margin of error represents the maximum and minimum accepted values of the quantity under consideration. The formula depends on whether the population standard deviation (σ) is known.

If σ is known and the sample is large (n > 30), the formula is: Margin of Error = z*(σ/√n).

If σ is unknown and the sample is small (n ≤ 30), the formula is: Margin of Error = t*(s/√n).

Here, z and t are critical values from z-statistics and t-statistics, associated with confidence levels (e.g., 90%, 95%, 99%). The term s/√n is called the standard error.

The Confidence Interval

A confidence interval is a range of values expected to contain a given quantity at a certain confidence level. For example, we might be 95% confident that the average height of 13-year-old girls is between 59 and 66 inches.

The formula is: x̄ ± Margin of Error

Example of Confidence Interval Calculation

Suppose we have a sample of 10 stock prices: 1.31, 1.36, 1.40, 1.27, 1.15, 0.99, 0.97, 0.88, 0.86, 0.80. We want to find the 95% confidence interval.

We have: Sample Mean (x̄) = 1.10, Sample Standard Deviation (s) = 0.23, n = 10. The critical value (t) for a 95% confidence level with 9 degrees of freedom is 2.26.

The margin of error is 2.26 * (0.23 / √10) ≈ 0.16.

The confidence interval is 1.10 ± 0.16, which is (0.94, 1.26).

This means we are 95% sure that the average stock price for the month lies between $0.94 and $1.26.