In mathematics, particularly in statistics, the term “range” is used to describe the spread of data. It’s a simple yet powerful concept that helps us understand the variability within a dataset. Essentially, the range tells us the distance between the highest and lowest values in a collection of numbers.
To find the range, you simply subtract the smallest value from the largest value in your dataset. This single number provides a quick snapshot of how much the data points are dispersed.
For example, consider the dataset: $left{5, 8, 2, 12, 3right}$. To find the range, we identify the highest value (12) and the lowest value (2). Subtracting the lowest from the highest, we get $12 – 2 = 10$. Therefore, the range of this dataset is 10.
But what does this range of 10 actually tell us? It indicates the total span of our data. A larger range suggests greater variability, meaning the data points are more spread out. Conversely, a smaller range implies less variability, with data points clustered closer together.
Range Definition and Formula
In mathematical terms, the range is formally defined as the difference between the maximum and minimum values in a dataset. It’s the most straightforward measure of dispersion, offering an initial understanding of data spread at a glance.
Let’s solidify this with another example: What is the range of the following set of numbers: $left{15, 22, 35, 10, 18right}$?
First, we identify:
- The largest value = 35
- The smallest value = 10
Then, we apply the range formula:
Range = Maximum Value – Minimum Value
Range = $35 – 10 = 25$
This range of 25 signifies that the largest number in the set is 25 units away from the smallest number. A larger range generally points to higher variability within the dataset.
How to Calculate Range: Step-by-Step Guide
Calculating the range is a straightforward process. Follow these simple steps to find the range of any dataset:
Step 1: Arrange the Data in Ascending Order
While not strictly necessary, arranging the numbers from lowest to highest makes it easier to identify the minimum and maximum values, especially in larger datasets.
Step 2: Identify the Lowest and Highest Values
Once sorted (or even by simply scanning the dataset), pinpoint the smallest (minimum) and largest (maximum) numbers.
Step 3: Subtract the Lowest Value from the Highest Value
Apply the formula: Range = Highest Value – Lowest Value. The result is the range of your dataset.
Example: Let’s calculate the range of ages of participants in a coding workshop:
| Age | 25 | 16 | 32 | 28 | 20 | 45 | 52 | 30 |
|---|
Following our steps:
| Age | 16 (Lowest) | 20 | 25 | 28 | 30 | 32 | 45 | 52 (Highest) |
|---|
Range = Highest Value – Lowest Value
Range = $52 – 16 = 36$
The range of ages in the workshop is 36 years. This indicates a considerable age gap among the participants.
Understanding Range in Different Contexts
The concept of range is versatile and can be applied in various mathematical and real-world scenarios. Let’s explore a few specific contexts:
Range Between Two Numbers
In the simplest case, if your dataset consists of only two numbers, the range is simply the absolute difference between them.
Example: Consider the set $left{7, 19right}$.
Range = $19 – 7 = 12$
The range is 12, representing the difference between these two numbers.
Comparing Data Sets with Range
Range becomes particularly useful when comparing the variability of two or more datasets, especially when they are related or measuring similar attributes.
Example: Let’s compare the daily sales figures for two coffee shops, Shop C and Shop D, over a week:
Shop C Sales (in $): 350, 380, 420, 420, 450, 460, 480
Shop D Sales (in $): 280, 350, 370, 370, 370, 380, 390
Range for Shop C = $480 – 350 = $130$
Range for Shop D = $390 – 280 = $110$
Comparing the ranges, Shop C has a slightly larger range ($130) than Shop D ($110). This suggests that Shop C’s daily sales are more variable compared to Shop D, even though both shops have sales within a similar ballpark.
Finding Range from Graphs
The range can also be visually determined from graphs like bar graphs or scatter plots, providing a quick interpretation of data spread directly from visual representations.
Example: Consider the bar graph below illustrating the average monthly rainfall (in inches) for a city.
By observing the bar graph:
- Highest rainfall is in August, approximately 4 inches.
- Lowest rainfall is in February, approximately 1 inch.
Range of rainfall = Highest rainfall – Lowest rainfall = $4 – 1 = 3$ inches.
The range of 3 inches tells us that the average monthly rainfall in this city varies by 3 inches throughout the year.
Advantages and Limitations of Range
Like any statistical measure, the range has its strengths and weaknesses. Understanding these helps in using it appropriately.
Merits of Range
- Simplicity and Ease of Calculation: Range is incredibly easy to calculate and understand, making it accessible even without advanced mathematical knowledge.
- Quick Indication of Variability: It provides an immediate sense of data spread, highlighting the total span of values.
- Useful for Initial Data Exploration: Range is helpful in preliminary data analysis to get a first impression of data dispersion.
- Application in Quality Control: In manufacturing and quality control, range can quickly indicate variations in product dimensions or other measurable attributes.
- Comparing Small Datasets: Range is suitable for comparing the variability of small datasets of similar size.
Limitations of Range
-
Sensitivity to Outliers: Range is highly affected by extreme values or outliers. A single very high or very low value can drastically inflate the range, misrepresenting the typical data spread.
Example: Consider the datasets:
Set 1: $left{4, 6, 8, 10right}$ Range = $10 – 4 = 6$
Set 2: $left{4, 6, 8, 10, 250right}$ Range = $250 – 4 = 246$
In Set 2, the outlier 250 dramatically increases the range, even though most data points are similar to Set 1.
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Ignores Most Data Points: Range only considers the two extreme values, disregarding the distribution of the data points in between. It doesn’t provide information about data clustering or central tendency.
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Limited Information about Data Structure: Range alone doesn’t reveal anything about the shape or structure of the data distribution. You can’t tell if data is evenly distributed or clustered from just the range.
Range in Functions (Algebra)
The concept of “range” also extends to functions in algebra. In the context of a function, the range refers to the set of all possible output values (y-values) that the function can produce. For a function $y = f(x)$, the range is the set of all possible values of $y$.
Example: For the function $f(x) = x^2$ where the domain (input values) is the set $left{-2, -1, 0, 1, 2right}$, let’s find the range.
By evaluating the function for each input value:
- $f(-2) = (-2)^2 = 4$
- $f(-1) = (-1)^2 = 1$
- $f(0) = (0)^2 = 0$
- $f(1) = (1)^2 = 1$
- $f(2) = (2)^2 = 4$
The set of output values is $left{4, 1, 0, 1, 4right}$. Therefore, the range of the function for this domain is $left{0, 1, 4right}$ (we only list unique values in a set).
Finding Range of a Function from a Graph
Graphically, the range of a function can be visualized along the y-axis. The range encompasses all the y-values that the graph of the function covers.
Consider these examples:
-
Graph extending upwards from y = 2: If a graph starts at $y = 2$ and extends upwards indefinitely, the range is $[2, infty)$. The square bracket indicates that 2 is included, and $infty$ represents infinity, meaning it goes on without bound.
-
Graph bounded between y = -3 and y = 3: If a graph is contained between $y = -3$ and $y = 3$, including these values, the range is $[-3, 3]$.
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Constant Function: For a constant function like $f(x) = 5$, the output is always 5, regardless of the input. Thus, the range is simply the set $left{5right}$.
Solved Examples on Range
Let’s work through a few more examples to solidify your understanding of range.
Example 1: Water Bottle Prices
During a road trip, Sarah noted the price of a water bottle at 7 different stores: $1.50, $2.25, $1.75, $2.50, $1.90, $2.00, $2.75. What is the range of water bottle prices?
Solution:
First, arrange the prices in ascending order: $1.50, $1.75, $1.90, $2.00, $2.25, $2.50, $2.75.
Lowest price = $1.50
Highest price = $2.75
Range = Highest price – Lowest price = $2.75 – $1.50 = $1.25
The range of water bottle prices is $1.25.
Example 2: Range of a Number Set
Find the range of the number set: $left{11, 18, 25, 14, 22, 9right}$.
Solution:
Arrange the set in ascending order: $left{9, 11, 14, 18, 22, 25right}$.
Lowest value = 9
Highest value = 25
Range = Highest value – Lowest value = $25 – 9 = 16$
The range of the given number set is 16.
Example 3: Range of Natural Numbers
What is the range of the set of natural numbers less than 20?
Solution:
Natural numbers less than 20 are: $left{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19right}$.
Lowest value = 1
Highest value = 19
Range = Highest value – Lowest value = $19 – 1 = 18$
The range of natural numbers less than 20 is 18.
Practice Questions on Range
Test your understanding with these practice questions:
Range in Math – Definition, Formula, Examples, Facts, FAQs
Attend this quiz & Test your knowledge.
1
Find the range: 200, 300, 950, 280, 175, 600.
775
750
800
725
CorrectIncorrect
Correct answer is: 775The largest value is 950. The smallest value is 175. Range $=$ Largest value $−$ Smallest value $= 950 ;−; 175 = 775$
2
The formula to calculate range is:
Minimum value − Maximum value
Largest value − Smallest value
$frac{Smallest; value; times ;Greatest; value}{2}$
Largest value $+$ Smallest value
CorrectIncorrect
Correct answer is: Largest value − Smallest valueRange $=$ Largest value$ ;−;$ Smallest value
3
Based on the data points 8, 3, 7, 12, 18, 30, 25, we can infer that:
Minimum value $= 3$
Maximum value $= 30$
Range $= 27$
All of the above
CorrectIncorrect
Correct answer is: All of the aboveMinimum value $= 3$ Maximum value $= 30$ Range $=$ Largest value $−$ Smallest value $= 27$
4
What is the range of a function $f(x) = -2$?
$left{2right}$
$left{-2right}$
$left{-1, 1right}$
$left{0right}$
CorrectIncorrect
Correct answer is: $left{-2right}$The function $f(x) = -2$ is a constant function. Thus, the range includes a single number, -2. Range $= left{-2right}$
Frequently Asked Questions about Range
What are quartiles in statistics?
Quartiles are values that divide a dataset into four equal parts. There are three quartiles: the first quartile ($Q_1$), the second quartile ($Q_2$, which is also the median), and the third quartile ($Q_3$). They are used to understand the distribution and spread of data.
What is the interquartile range?
The interquartile range (IQR) is a measure of statistical dispersion, representing the range of the middle 50% of the data. It is calculated as the difference between the third quartile ($Q_3$) and the first quartile ($Q_1$): $IQR = Q_3 – Q_1$. IQR is less sensitive to outliers than the overall range.
What is the key difference between mean and range?
Mean (average) and range are both descriptive statistics but measure different aspects of data. Mean is a measure of central tendency, indicating the average value of a dataset. It uses all data points to calculate a central value. Range, on the other hand, is a measure of dispersion, indicating the spread of the data by focusing only on the difference between the maximum and minimum values. Mean is influenced by all values, while range is only influenced by the extremes.
Conclusion
The range is a fundamental statistical concept that provides a quick and easy way to understand the spread or variability within a dataset. While it has limitations, especially concerning outliers and the lack of detail about data distribution, its simplicity and ease of calculation make it a valuable tool for initial data analysis and comparison, and for understanding the spread in various contexts, including functions in algebra. Understanding range is a crucial stepping stone to grasping more complex statistical measures and data analysis techniques.
