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Flashcards: Summarizing numerical data sets

Consider the following data set:

$\left \{ 1, 4, 5, 8, 8, 9, 10, 11, x, x \right \}$ $\displaystyle \left \{ 1, 4, 5, 8, 8, 9, 10, 11, x, x \right \}$

If $x \ne 8$ $\displaystyle x \ne 8$ , which is true of the mode or modes of the data set?

The set has one mode, 8.

The set has two modes, 8 and $\displaystyle x$ .

The set has one mode, $\displaystyle x$ .

The set has no mode.

The question cannot be answered from the information given.

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Flashcards: Summarizing numerical data sets

Consider the following data set:

$\left \{ 1, 4, 5, 8, 8, 9, 10, 11, x, x \right \}$ $\displaystyle \left \{ 1, 4, 5, 8, 8, 9, 10, 11, x, x \right \}$

If $x \ne 8$ $\displaystyle x \ne 8$ , which is true of the mode or modes of the data set?

The set has one mode, 8.

The set has two modes, 8 and $\displaystyle x$ .

The set has one mode, $\displaystyle x$ .

The set has no mode.

The question cannot be answered from the information given.

Report an Error Copyright Notice

Flashcards: Summarizing numerical data sets

Consider the following data set:

$\left \{ 1, 4, 5, 8, 8, 9, 10, 11, x, x \right \}$ $\displaystyle \left \{ 1, 4, 5, 8, 8, 9, 10, 11, x, x \right \}$

If $x \ne 8$ $\displaystyle x \ne 8$ , which is true of the mode or modes of the data set?

Your answer:

The set has one mode, 8.

Your answer:

The set has two modes, 8 and $\displaystyle x$ .

Your answer:

The set has one mode, $\displaystyle x$ .

Your answer:

The set has no mode.

Correct answer:

The question cannot be answered from the information given.

Explanation:

The mode of a data set is the value that occurs most frequently in the set. If two or more values tie for most frequently occurring value, then the set has multiple modes.

We show that the question of the modes of the data set cannot be answered with certainty as follows:

Suppose $\displaystyle x$ assumes a value in the data set - for example, let x= 1 $\displaystyle x= 1$ . the data set is

$\left \{ 1,1, 1, 4, 5, 8, 8, 9, 10, 11 \right \}$ $\displaystyle \left \{ 1,1, 1, 4, 5, 8, 8, 9, 10, 11 \right \}$ ,

in which the most frequently occurring value is 1, which appears . This makes 1, $\displaystyle x$ , the mode.

Now, suppose $\displaystyle x$ assumes a value not already in the data set; for example, let x= 2 $\displaystyle x= 2$ . The data set becomes

$\left \{ 1,2,2 , 4, 5, 8, 8, 9, 10, 11 \right \}$ $\displaystyle \left \{ 1,2,2 , 4, 5, 8, 8, 9, 10, 11 \right \}$ ,

in which two values, 2 (the $\displaystyle x$ ) and 8 each occur twice.

Therefore, from the information given, the mode(s) cannot be determined with any certainty.

Flashcards: Summarizing numerical data sets

Consider the data set

$\left \{ 10, 16, 16, 18, 21, 25, x, 2x \right \}$ $\displaystyle \left \{ 10, 16, 16, 18, 21, 25, x, 2x \right \}$ ,

where $\displaystyle x$ is a prime integer.

How many possible values of $\displaystyle x$ make the set bimodal?

One

Three

Eight

Five

Four

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Flashcards: Summarizing numerical data sets

Consider the data set

$\left \{ 10, 16, 16, 18, 21, 25, x, 2x \right \}$ $\displaystyle \left \{ 10, 16, 16, 18, 21, 25, x, 2x \right \}$ ,

where $\displaystyle x$ is a prime integer.

How many possible values of $\displaystyle x$ make the set bimodal?

Your answer:

Three

Your answer:

Eight

Your answer:

Five

Your answer:

Four

Correct answer:

One

Explanation:

The mode of a data set is the value that occurs most frequently in the set. If two values tie for most frequently occurring value, then the set has two modes - it is bimodal.

The value 16 already occurs twice in the data set. For the set to be bimodal, one of the following must happen:

Case 1: $\displaystyle x= 2x = 0$ , which is not possible, since 0 is not considered prime or composite.

Case 2: $\displaystyle x$ must be equal to one of the other four known values, 10, 18, 21, or 25. however, $\displaystyle x$ is given to be prime - that is, to have only two factors, 1 and itself. Each of 10, 18, 21, and 25 is composite, having factors not equal to 1 or itself, so $\displaystyle x$ cannot assume any of these values.

Case 3: $\displaystyle 2x$ must be equal to one of the other four known values, 10, 18, 21, or 25. Then $\displaystyle x$ must be equal to half of the selected number:

$2x = 10 \Rightarrow x= 5$ $\displaystyle 2x = 10 \Rightarrow x= 5$

$2x = 18 \Rightarrow x= 9$ $\displaystyle 2x = 18 \Rightarrow x= 9$

$2x = 21 \Rightarrow x= 10.5$ $\displaystyle 2x = 21 \Rightarrow x= 10.5$

$2x = 25 \Rightarrow x= 12.5$ $\displaystyle 2x = 25 \Rightarrow x= 12.5$

Only in one case does it hold that $\displaystyle x$ is a prime integer.

The correct response is one - x= 5 $\displaystyle x= 5$ .

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