Frequency Distribution Calculator

What is a Frequency Distribution?

A frequency distribution is a tabular or graphical representation of data that shows the number of times each value occurs in a dataset. It provides a summary of the distribution of values in a sample.

Key Concepts and Formulas

1. Frequency (\(f\)): The number of times a particular value appears in a dataset.

2. Mean (\(\bar{x}\)): \[\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}\] Where \(x_i\) are individual values and \(n\) is the number of values.

3. Standard Deviation (\(s\)): \[s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}}\] This measures the spread of the data around the mean.

4. Standard Error (\(SE\)): \[SE = \frac{s}{\sqrt{n}}\] This estimates the standard deviation of the sampling distribution of the mean.

Calculation Steps

1. Count the frequency of each unique value in the dataset.
2. Calculate the mean by summing all values and dividing by the count.
3. Calculate the squared differences from the mean for each value.
4. Find the average of these squared differences to get the variance.
5. Take the square root of the variance to get the standard deviation.
6. Divide the standard deviation by the square root of the sample size to get the standard error.

Example Calculation

Let's calculate for the dataset: 2, 4, 4, 4, 5, 5, 7, 9

1. Frequency Distribution:
   • 2: 1
   • 4: 3
   • 5: 2
   • 7: 1
   • 9: 1
2. Mean: \(\bar{x} = \frac{2 + 4 + 4 + 4 + 5 + 5 + 7 + 9}{8} = 5\)
3. Squared differences: \((2-5)^2, (4-5)^2, ..., (9-5)^2\)
4. Variance: \(s^2 = \frac{9 + 1 + 1 + 1 + 0 + 0 + 4 + 16}{8} = 4\)
5. Standard Deviation: \(s = \sqrt{4} = 2\)
6. Standard Error: \(SE = \frac{2}{\sqrt{8}} \approx 0.707\)

Visual Representation

This histogram represents the frequency distribution of the example dataset. The height of each bar corresponds to the frequency of each value. The red dashed line indicates the mean (5).