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Statistics and Probability - Measures of Spread (Range)

Grade 7IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The Range is a measure of spread (or dispersion) that describes the difference between the highest and lowest values in a data set. It provides a quick way to see how 'stretched' the data is. Visually, on a number line, the range represents the total distance between the two outermost data points.

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The Maximum is the largest value in the data set, while the Minimum is the smallest value. Identifying these two values is the first step in calculating the range. In a dot plot, the minimum is the furthest dot to the left, and the maximum is the furthest dot to the right.

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The range is used to describe Consistency. A small range indicates that the data points are clustered closely together, meaning the data is consistent. A large range suggests the data is widely spread out and more variable. For example, a student with test scores with a range of 55 is more consistent than a student with a range of 3030.

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The range is highly sensitive to Outliers. An outlier is a data point that is significantly higher or lower than the rest of the values. Since the range only uses the two extreme values, one single outlier can make the spread of the data look much larger than it actually is for the majority of the group.

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To accurately find the range, it is best practice to Order the Data from least to greatest first. This visual organization helps ensure that you do not overlook a very small or very large number hidden in the middle of a disorganized list.

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The range is a Single Value, not a pair of numbers. While students often say 'the range is from 1010 to 5050', in mathematics, the range is the result of the subtraction: 50−10=4050 - 10 = 40. It always carries the same units as the original data (e.g., if data is in kilograms, the range is in kilograms).

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On a Box Plot (or Box-and-Whisker Plot), the range is represented by the total length of the diagram from the tip of the left whisker to the tip of the right whisker. This visual 'span' shows the full extent of the data's distribution.

📐Formulae

Range=Maximum Value−Minimum Value\text{Range} = \text{Maximum Value} - \text{Minimum Value}

Max=largest number in the set\text{Max} = \text{largest number in the set}

Min=smallest number in the set\text{Min} = \text{smallest number in the set}

💡Examples

Problem 1:

A group of 7 students recorded their heights in centimeters: 152,148,165,150,155,147,161152, 148, 165, 150, 155, 147, 161. Calculate the range of their heights.

Solution:

  1. First, arrange the data in ascending order: 147,148,150,152,155,161,165147, 148, 150, 152, 155, 161, 165
  2. Identify the minimum value: Min=147\text{Min} = 147 cm
  3. Identify the maximum value: Max=165\text{Max} = 165 cm
  4. Apply the formula: Range=165−147\text{Range} = 165 - 147
  5. Range=18\text{Range} = 18 cm

Explanation:

By ordering the numbers first, we clearly see that the shortest student is 147147 cm and the tallest is 165165 cm. Subtracting these gives a range of 1818 cm, representing the total spread of heights.

Problem 2:

Compare the consistency of two golfers based on their scores over 5 rounds. Golfer A: 72,71,73,72,7272, 71, 73, 72, 72. Golfer B: 65,80,70,75,7065, 80, 70, 75, 70. Which golfer is more consistent?

Solution:

  1. Calculate Range for Golfer A: Max=73,Min=71\text{Max} = 73, \text{Min} = 71. RangeA=73−71=2\text{Range}_A = 73 - 71 = 2
  2. Calculate Range for Golfer B: Max=80,Min=65\text{Max} = 80, \text{Min} = 65. RangeB=80−65=15\text{Range}_B = 80 - 65 = 15
  3. Compare the ranges: 2<152 < 15.

Explanation:

Golfer A is much more consistent because their range (22) is significantly smaller than Golfer B's range (1515). This means Golfer A's scores are clustered tightly together, while Golfer B's performance varies widely from round to round.