Mode & Range Calculator
A Mode and Range Calculator determines the mode (the most frequently occurring value) and the range (the spread between maximum and minimum values) of any dataset, with a complete sorted frequency distribution.
- Step 1: Parse and sort the dataset of 7 values in ascending order: [72.00, 72.00, 72.00, 75.00, 78.00, 80.00, 85.00].
- Step 2: Identify the minimum value: Min = 72.00.
- Step 3: Identify the maximum value: Max = 85.00.
- Step 4: Compute the Range: Max - Min = 85.00 - 72.00 = 13.00.
- Step 5: Tally the frequency of each unique number:
- • Value 72.00: appears 3 times.
- • Value 75.00: appears 1 time.
- • Value 78.00: appears 1 time.
- • Value 80.00: appears 1 time.
- • Value 85.00: appears 1 time.
- Step 6: Determine the Mode: The highest frequency is 3 occurrences. The mode is: 72.00 (unimodal).
How Mode and Range are Calculated
While the average (mean) and median identify the center of a dataset, the mode identifies its most popular values, and the range quantifies its total dispersion.
To find the mode, organize the data into an ordered frequency table. The value with the greatest tally is the mode. If two values tie for highest frequency, the dataset is bimodal; if all numbers occur an equal number of times, there is no mode.
To find the range, subtract the lowest numerical value from the highest numerical value.
Worked Example: Daily High Temperatures in a City
A meteorologist records the following daily high temperatures (°F) over a seven-day week: 72, 75, 78, 72, 80, 72, 85. What are the range and mode of the weekly temperatures?
- Sort the temperatures in ascending order: [72, 72, 72, 75, 78, 80, 85].
- Identify the minimum temperature: Min = 72°F.
- Identify the maximum temperature: Max = 85°F.
- Compute the Range: Max − Min = 85 − 72 = 13°F.
- Tally the frequencies of each unique value:
- • 72°F appears 3 times.
- • 75°F, 78°F, 80°F, and 85°F each appear 1 time.
- The highest frequency is 3 occurrences for 72°F.
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Statistical Comparison: Mode, Range, Mean, and Median Across Sample Distributions
| Dataset | Sorted Order | Range (Max - Min) | Mode | Mode Type |
|---|---|---|---|---|
| 4, 8, 8, 12, 15 | 4, 8, 8, 12, 15 | 11 | 8 | Unimodal |
| 10, 10, 20, 30, 30 | 10, 10, 20, 30, 30 | 20 | 10, 30 | Bimodal |
| 5, 12, 19, 27 | 5, 12, 19, 27 | 22 | No mode | None (All Unique) |
| 7, 7, 14, 14, 21, 21 | 7, 7, 14, 14, 21, 21 | 14 | No mode | None (Uniform Frequency) |
| -10, -5, 0, 5, 5 | -10, -5, 0, 5, 5 | 15 | 5 | Unimodal |
| 100, 100, 100, 100 | 100, 100, 100, 100 | 0 | 100 | Unimodal (Uniform Single Value) |
Common Mistakes With Mode and Range
If a dataset contains [2, 4, 6, 8], each number occurs once. Stating that the mode is "0" is a serious mathematical error unless the number 0 actually appears in the dataset. When all numbers occur with equal frequency, the dataset has "No mode".
Range is a non-negative measure of statistical dispersion. It is always calculated as Maximum minus Minimum. Calculating Min − Max produces a negative number, which is mathematically invalid for range.
When the minimum value is negative, subtracting a negative number becomes addition: if Max is 15 and Min is −10, Range = 15 − (−10) = 15 + 10 = 25, not 5.
Frequently Asked Questions
- What is the mode in statistics?
- The mode is the numerical value that appears most frequently in a dataset. A dataset can have one mode (unimodal), two modes (bimodal), multiple modes (multimodal), or no mode at all.
- Can a dataset have more than one mode?
- Yes. If two or more distinct values tie for the highest frequency of occurrences, the dataset is multimodal (e.g. bimodal if two values tie).
- What is the range of a dataset?
- The range is the difference between the largest number (maximum) and the smallest number (minimum) in a dataset: Range = Max − Min.
- Why is range sensitive to outliers?
- Because range depends entirely on the two extreme values (the highest and lowest numbers), a single freak measurement or outlier drastically inflates the reported range.
- When is mode preferred over mean or median?
- Mode is especially useful for categorical data (e.g. most popular shoe size, most common voting preference) or discrete counts where the mathematical average would produce a non-existent fractional category.