Standard Deviation Calculator Frequency Table

Standard Deviation Formula:

\[ \sigma = \sqrt{\frac{\sum f_i (x_i - \mu)^2}{\sum f_i}} \]

Mean (μ):

Variance:

Standard Deviation (σ):

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1. What is a Standard Deviation Calculator for Frequency Tables?

Definition: This calculator computes the standard deviation from grouped data presented in a frequency table.

Purpose: It helps statisticians, researchers, and students analyze the dispersion of data points when working with frequency distributions.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \sigma = \sqrt{\frac{\sum f_i (x_i - \mu)^2}{\sum f_i}} \]

Where:

\( \sigma \) — Standard deviation (measures dispersion)
\( f_i \) — Frequency of each data point
\( x_i \) — Value of each data point
\( \mu \) — Mean of the data set

Explanation: The calculator first computes the weighted mean, then calculates how far each point deviates from the mean (squared and weighted by frequency), and finally takes the square root of the average of these squared deviations.

3. Importance of Standard Deviation Calculation

Details: Standard deviation quantifies the amount of variation in a data set. A low standard deviation indicates data points tend to be close to the mean, while a high standard deviation indicates data points are spread out.

4. Using the Calculator

Tips:

Enter each unique data value (x_i) and its corresponding frequency (f_i)
Add or remove rows as needed for your data set
All frequencies must be positive integers
All data points must be numeric values

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between population and sample standard deviation?
A: For population SD, divide by N (total frequency). For sample SD, divide by N-1. This calculator computes population SD.

Q2: How do I interpret the standard deviation value?
A: Higher values mean more spread in the data. About 68% of values lie within ±1σ of the mean in normal distributions.

Q3: What if my frequency table has class intervals instead of exact values?
A: Use class midpoints as x_i values in the calculator.

Q4: Why do we square the deviations?
A: Squaring eliminates negative values and gives more weight to larger deviations.

Q5: Can I use this for probability distributions?
A: Yes, if you enter the possible outcomes as x_i and their probabilities as f_i (scaled to integers).