Advanced Statistics Calculator
Instructions:
- Enter a list of numbers separated by commas into the text area below (e.g., 1, 2, 3, 4, 5).
- Click the "Calculate Statistics" button.
- The results for Mean, Median, Mode, Range, Variance, Standard Deviation, Interquartile Range (IQR), and Skewness will be displayed.
- For each metric, you can click "Show Details" to view the formula, theory, and a step-by-step calculation. Click "Hide Details" to collapse it.
- Click "Clear Data" to reset the input and results.
Results:
Mean: -
Formula:
Sample Mean (x̄) = Σxᵢ / n
Theory:
The mean, or average, is the sum of all values divided by the number of values. It represents a typical value or the center of a dataset. It's sensitive to outliers.
Step-by-step Calculation:
Median: -
Formula:
1. Sort the data: x₁, x₂, ..., xₙ.
2. If n is odd, Median = x(n+1)/2.
3. If n is even, Median = (xn/2 + x(n/2)+1) / 2.
Theory:
The median is the middle value of a dataset that has been sorted in ascending order. If there's an even number of observations, the median is the average of the two middle values. It's less affected by outliers than the mean.
Step-by-step Calculation:
Mode: -
Formula:
The value(s) that appear most frequently in the dataset.
Theory:
The mode is the value that occurs most often in a dataset. A dataset can have one mode (unimodal), more than one mode (multimodal), or no mode if all values occur with the same frequency.
Step-by-step Calculation:
Range: -
Formula:
Range = Maximum Value - Minimum Value
Theory:
The range is the difference between the largest and smallest values in a dataset. It provides a simple measure of spread but is highly influenced by outliers.
Step-by-step Calculation:
Sample Variance (s²): -
Formula:
Sample Variance (s²) = Σ(xᵢ - x̄)² / (n - 1)
Theory:
Variance measures how spread out the numbers in a dataset are around their mean. A small variance indicates that the data points tend to be very close to the mean, while a large variance indicates that the data points are spread out over a wider range of values. We use (n-1) for sample variance to provide an unbiased estimate of the population variance.
Step-by-step Calculation:
Sample Standard Deviation (s): -
Formula:
Sample Standard Deviation (s) = √s² = √[Σ(xᵢ - x̄)² / (n - 1)]
Theory:
The standard deviation is the square root of the variance. It measures the average amount of variability or dispersion in a dataset. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.
Step-by-step Calculation:
Interquartile Range (IQR): - (Q1: -, Q3: -)
Formula:
IQR = Q3 - Q1
Q1 (First Quartile): Value at the 25th percentile.
Q3 (Third Quartile): Value at the 75th percentile.
Theory:
The IQR is a measure of statistical dispersion, being equal to the difference between the 75th (Q3) and 25th (Q1) percentiles. It represents the range of the middle 50% of the data and is less sensitive to outliers than the range.
Step-by-step Calculation:
Sample Skewness (g₁): -
Formula (Adjusted Fisher-Pearson standardized moment coefficient):
g₁ = [n / ((n-1)(n-2))] * Σ((xᵢ - x̄) / s)³
(Alternative simpler formula: Pearson's Second Coefficient of Skewness = 3 * (Mean - Median) / Standard Deviation, often used as a rough guide)
Theory:
Skewness measures the asymmetry of the probability distribution of a real-valued random variable about its mean.
- If skewness is 0, the data are perfectly symmetrical.
- If skewness is negative (g₁ < 0), the data are skewed left (tail on the left side is longer).
- If skewness is positive (g₁ > 0), the data are skewed right (tail on the right side is longer).
Values between -0.5 and 0.5 are considered fairly symmetrical.