Calculate Min, Max, and Range
Online calculator for descriptive statistics: minimum, maximum, range, count, mean, and interquartile range
Statistics Calculator
Descriptive Statistics
Calculate the important measures of location and dispersion of a data series: minimum, maximum, range, count, mean, and interquartile range.
Data Distribution
Visualization of statistical measures
Minimum, maximum, range, and interquartile range show the data distribution
● Minimum ● Quartiles ● Median ● Maximum
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Descriptive Statistics - Fundamentals
Descriptive statistics are fundamental measures for characterizing datasets:
- Minimum (Min): The smallest value in the data series
- Maximum (Max): The largest value in the data series
- Range: The difference between maximum and minimum (Max - Min)
- Count (n): The total number of data points
- Mean (Ø): The arithmetic average of all values
- Interquartile Range (IQR): The distance between the 3rd and 1st quartile (Q3 - Q1)
Calculating the Measures
Step-by-step explanation for calculating the measures:
Minimum
- The smallest value of the data series
- Min = min(x₁, x₂, ..., xₙ)
- Example: For [3, 1, 7, 5] → Min = 1
Maximum
- The largest value of the data series
- Max = max(x₁, x₂, ..., xₙ)
- Example: For [3, 1, 7, 5] → Max = 7
Range
- The difference between Max and Min
- Range = Max - Min
- Example: For [3, 1, 7, 5] → Range = 7 - 1 = 6
Count
- The total number of data points
- n = Length of the data series
- Example: For [3, 1, 7, 5] → n = 4
Mean
- The arithmetic average
- Ø = (x₁ + x₂ + ... + xₙ) / n
- Example: For [3, 1, 7, 5] → Ø = 16/4 = 4
Interquartile Range
- Difference between 3rd and 1st quartile
- IQR = Q3 - Q1
- Contains the middle 50% of the data
Applications and Significance
These statistical measures are used in many practical applications:
Exploratory Data Analysis
- Quick overview of data distribution
- Detection of outliers (outside Min-Max)
- Boxplot visualization
- Data quality checks
Practical Applications
- Quality control in manufacturing
- Financial risk analysis
- Climate data analysis (Min/Max temperatures)
- Measurement accuracy analysis
Formulas and Definitions
Minimum
The smallest value in the data series
Maximum
The largest value in the data series
Range
The difference between maximum and minimum
Arithmetic Mean
The average of all values
Interquartile Range (IQR)
The middle 50% of the data (between 25th and 75th percentile)
Five-Number Summary
Complete summary of the data distribution
Symbol Explanations
| \(x_i\) | i-th data point |
| \(n\) | Number of data points |
| \(Q_1\) | 1st quartile (25%) |
| \(Q_3\) | 3rd quartile (75%) |
| \(\overline{x}\) | Mean |
| \(R\) | Range |
Example Calculations
Example 1: Test Scores
Calculate: All descriptive statistics
1. Sort Data
Ascending sorting required for Min/Max
2. Determine Min & Max
Min is the 1st value, Max is the last value
3. Count & Mean
Average of all values
4. Quartiles & IQR
The middle 50% of the data
Example 2: Daily Temperatures for a Week (in °C)
Calculate: Temperature range and average temperature
Results
Minimum
18°C
Maximum
25°C
Range
7°C
Mean
21.71°C
Number of Days
7
Example 3: Data Set Comparison
Data Set A: 10, 20, 30, 40, 50 → Min=10, Max=50, Range=40, Ø=30
Data Set B: 25, 26, 30, 34, 35 → Min=25, Max=35, Range=10, Ø=30
Both have the same mean, but different ranges! Data Set A is much more widely distributed.
Mathematical Foundations
Descriptive statistics form the basis of exploratory data analysis and are essential for understanding data distributions.
Properties of Measures of Location
- Invariance: Min and Max are not invariant under monotonic transformations, but change predictably
- Robustness: Min and Max are not robust to outliers (one extreme value changes them)
- Simplicity: These measures are intuitively understandable and easy to calculate
- Information: The range provides information about the variability of the data
Five-Number Summary
The Five-Number Summary is a concept for completely characterizing a distribution:
The Five Numbers
- Minimum: 0th percentile
- Q₁: 25th percentile (lower quartile)
- Median: 50th percentile (2nd quartile)
- Q₃: 75th percentile (upper quartile)
- Maximum: 100th percentile
Boxplot Visualization
The Five-Number Summary is typically visualized as a boxplot: The box shows the middle 50% (IQR), the line in the box is the median, and the whiskers show Min and Max.
Quartiles and Percentiles
Quartiles are special cases of percentiles:
- Q₁ (25th percentile): 25% of the data lie below it
- Q₂ (50th percentile) = Median: 50% of the data lie below it
- Q₃ (75th percentile): 75% of the data lie below it
- IQR = Q₃ - Q₁: The range of the middle 50% of the data
- Outlier thresholds: Values < Q₁ - 1.5·IQR or > Q₃ + 1.5·IQR are often considered outliers
Practical Significance
Data Quality Checking
Min and Max show the range of values and help identify impossible values (e.g., negative temperatures where none are possible).
Outlier Detection
The interquartile range is the basis for outlier detection using the 1.5·IQR rule, one of the most robust methods for detecting anomalies.
Summary
These fundamental statistical measures form the basis of data analysis. They are easy to understand, quick to calculate, and provide a quick overview of data structure and distribution. Together with the boxplot concept, they form a fundamental method of exploratory data analysis (EDA).
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