Question

Which is larger for this set of numbers, the mode or the range? $$ \\{1,1,4,5,12,71\\} $$ (A) Mode OR (B) Range

Short answer

The range is larger (B).

Step-by-step solution

Determine the Mode

The mode of a dataset is the number that appears most frequently. In the set \( \{1, 1, 4, 5, 12, 71\} \), the number 1 appears twice, while all other numbers appear only once. Therefore, the mode of this set is 1.

Calculate the Range

The range of a dataset is calculated by subtracting the smallest number in the set from the largest number. In this dataset, the smallest number is 1 and the largest number is 71. \(Range = 71 - 1 = 70\)

Compare the Mode and the Range

Now, compare the mode and the range. We have the mode as 1 and the range as 70. Since 70 is larger than 1, the range is larger.

Key concepts

Understanding the Mode

Mode is a simple yet important concept in statistics, especially useful when analyzing data in PSAT Math problems. In any set of numbers, the mode is the value that occurs most frequently. It provides insight into the most common or typical value of your dataset.

• If one number appears more often than others, that's your mode.
• If no number repeats, there's no mode.
• There can also be more than one mode if multiple numbers appear with the same maximum frequency—this is called a multimodal set.

Let's look at an example: in the set \( \{1, 1, 4, 5, 12, 71\} \), the number 1 appears twice, whereas all other numbers occur just once. Thus, the mode of the set is 1.
Mode can be especially helpful in surveys or tests where you want to find the most common response or score. It gives invaluable insight for decision-making based on the frequency distribution.

Exploring the Range

The range of a set of numbers represents the difference between the largest and smallest numbers. It's a measure of how spread out the numbers are. Knowing the range helps to understand the variability or dispersion in your data.

• To find the range, subtract the smallest number from the largest number in your set.
• A small range indicates that the numbers are closely packed together.
• A large range indicates that there's significant variability between the smallest and largest values.

For example, in the number set \( \{1, 1, 4, 5, 12, 71\} \), the smallest number is 1 and the largest is 71. So, the range is calculated as \(71 - 1 = 70\).
While the range provides a quick insight into data variability, it's important to remember that it only considers the two extreme values and not every number in between. In datasets with outliers, the range can give a misleading impression of general data dispersion.

Delving into Number Sets

Number sets are everywhere in math, and understanding them is crucial for tackling PSAT Math problems effectively. A number set is simply a collection of numbers that you can analyze and evaluate using different statistical measures such as mean, median, mode, and range.

• A set can contain any type of number, including whole numbers, fractions, or even negative numbers.
• Each number set can have particular properties or unique features, such as having a distinct range or specific mode.
• A set might be small or large depending on the context of the problem.

Consider the number set \( \{1, 1, 4, 5, 12, 71\} \). This set is specific because it displays repetition with the number 1 being the mode and spans a wide range from 1 to 71.
Recognizing how to classify and handle different number sets can sharpen your mathematical understanding and prepare you for effectively solving various questions on tests like the PSAT. Developing an intuitive feel for these sets helps in knowing which statistical tools to utilize for answering specific questions accurately.