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Chapter 2: Problem 5

a. Explain the difference between a discrete variable and a continuous variable. b. Give an example of each type.

Short Answer

Expert verified
A discrete variable has specific individual values; a continuous variable can take any value in a range. Examples: number of students (discrete), height of students (continuous).

Step by step solution

01

Define a Discrete Variable

A discrete variable is a type of quantitative variable that can take on a countable number of distinct values. These values are typically whole numbers, and no intermediate values between them are possible. Discrete variables often represent counts or certain categories. For instance, the number of students in a classroom is a discrete variable because you can count the exact number of students, and fractional students do not exist.
02

Define a Continuous Variable

A continuous variable, unlike a discrete variable, can take on an infinite number of values within a given range. These variables can assume any value along a continuum, including fractions and decimals. Continuous variables are often measurements and can represent detailed numerical differences. For example, the height of a student is a continuous variable because it can be any value within a reasonable range and can include fractions of a unit (e.g., 170.5 cm).
03

Provide Example of a Discrete Variable

An example of a discrete variable is the number of cars in a parking lot. This variable is discrete because each count is distinct (e.g., 1, 2, 3 cars), and fractional cars do not make sense.
04

Provide Example of a Continuous Variable

An example of a continuous variable is the time it takes to run a race. This is a continuous variable because it can be any positive real number, including fractions of a second.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quantitative Variables
In statistics, quantitative variables are variables that represent numeric measurements or counts. They differ from qualitative variables, which describe categories or characteristics without inherent numeric values. Quantitative variables are essential in statistical analysis because they convey measurable information. They can be divided into two main types: discrete and continuous variables.
  • Quantitative variables allow us to conduct mathematical operations like addition, subtraction, and averaging.
  • They provide a way to quantify phenomena and are the basis for statistical calculations and data representation.
Understanding the distinction between different quantitative variables helps in choosing the right statistical methods for analysis.
Discrete Variables
Discrete variables are a type of quantitative variable that can only take on specific values, which are countable. These variables typically represent counts of things or categories, meaning they are often whole numbers and do not have any intermediate values.
  • For example, the number of students in a classroom is a discrete variable. You can count each student, but you can't have a fraction of a student.
  • Voting results in an election are also discrete because they represent integer counts of votes.
Discrete variables are often visualized using bar charts since these highlight the different distinct categories or possibilities. When working with discrete variables, it is essential to remember they are finite and do not include fractions or decimals, making them relatively straightforward to analyze in certain contexts.
Continuous Variables
Continuous variables, unlike discrete variables, can take on an infinite number of possible values within a given range. These are often measurements and can represent precise numerical differences.
  • Height and weight are classic examples of continuous variables since they can be measured to various degrees of precision, allowing for values including decimals.
  • The amount of time elapsed during an event is another example, where the time could be measured to fractions of a second.
Continuous variables are usually represented through histograms or line graphs, which help to show the distribution of values over the numeric range. In dealing with continuous variables, understanding how they span a continuum is crucial; this quality allows us to apply techniques like calculus to find more detailed statistical measures.

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Most popular questions from this chapter

If the largest observation is less than 1 standard deviation above the mean, then the distribution tends to be skewed to the left. If the smallest observation is less than 1 standard deviation below the mean, then the distribution tends to be skewed to the right. A professor examined the results of the first exam given in her statistics class. The scores were $$\begin{array}{llllllll} 35 & 59 & 70 & 73 & 75 & 81 & 84 & 86 \end{array}$$ The mean and standard deviation are 70.4 and 16.7 . Using these, determine whether the distribution is either left or right skewed. Construct a dot plot to check.

Bad statistic \(\quad\) A teacher summarizes grades on an exam by \(\operatorname{Min}=26, \mathrm{Q} 1=67, \mathrm{Q} 2=80, \mathrm{Q} 3=87, \operatorname{Max}=100\) Mean \(=76,\) Mode \(=100,\) Standard deviation \(=76\) \(\mathrm{IQR}=20\) She incorrectly recorded one of these. Which one do you think it was? Why?

The Human Development Report 2013, published by the United Nations, showed life expectancies by country. For Western Europe, some values reported were Austria \(81,\) Belgium \(80,\) Denmark \(80,\) Finland 81 , France 83 , Germany \(81,\) Greece \(81,\) Ireland 81 , Italy 83, Netherlands 81 , Norway 81 , Portugal 80 , Spain 82 , Sweden \(82,\) Switzerland \(83 .\) For Africa, some values reported were Botswana \(47,\) Dem. Rep. Congo \(50,\) Angola \(51,\) Zambia \(57,\) Zimbabwe 58 , Malawi 55 , Nigeria \(52,\) Rwanda 63 , Uganda 59 , Kenya 61 , Mali 55 , South Africa 56 , Madagascar 64 , Senegal \(63,\) Sudan \(62,\) Ghana \(61 .\) a. Which group (Western Europe or Africa) of life expectancies do you think has the larger standard deviation? Why? b. Find the standard deviation for each group. Compare them to illustrate that \(s\) is larger for the group that shows more variability from the mean.

The standard deviation is the most popular measure of variability from the mean. It uses squared deviations because the ordinary deviations sum to zero. An alternative measure is the mean absolute deviation, \(\Sigma|x-\bar{x}| / n\) a. Explain why greater variability tends to result in larger values of this measure. b. Would the MAD be more, or less, resistant than the standard deviation? Explain.

According to a recent report from the U.S. National Center for Health Statistics, females between 25 and 34 years of age have a bell-shaped distribution for height, with mean of 65 inches and standard deviation of 3.5 inches. a. Give an interval within which about \(95 \%\) of the heights fall. b. What is the height for a female who is 3 standard deviations below the mean? Would this be a rather unusual height? Why?
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