AP Statistics : Univariate Data

Study concepts, example questions & explanations for AP Statistics

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Example Questions

Example Question #1 : Data

Six homes are for sale and have the following dollar values in thousands of dollars: 

535

155

305

720

315

214

What is the median value of the six homes?

Possible Answers:

\(\displaystyle 374\)

\(\displaystyle 310\)

\(\displaystyle 395\)

\(\displaystyle 315\)

\(\displaystyle 305\)

Correct answer:

\(\displaystyle 310\)

Explanation:

The median is determined by ordering the values in the group from least to greatest: 155, 214, 305, 315, 535, 720.  The value directly in the middle is the median.  For instance, if there are five numbers, the third is the median.  Here, we have an even number of values so there is no value directly in the middle.  To find the median when there is no middle-most value, find the average of the two middle values.  The mean is determined by adding the two values and dividing by two (the number in the group):

\(\displaystyle 305 + 315 = 620; 620 / 2 = 310\)

Example Question #1 : Data

Find the median of the set.

\(\displaystyle 1,2,3,4,5,6,7,8,9,10,11\)

Possible Answers:

\(\displaystyle 10\)

\(\displaystyle 7\)

\(\displaystyle 3\)

\(\displaystyle 6\)

Correct answer:

\(\displaystyle 6\)

Explanation:

The median is the middle value of the set in increasing order.

In this set of 11 entries, the median is the 6th entry of the set in increasing order, or 6.

Example Question #1 : How To Find The Median For A Set Of Data

Find the median of the set

\(\displaystyle 2,4,6,8,10,12\)

Possible Answers:

\(\displaystyle 10\)

\(\displaystyle 8\)

\(\displaystyle 6\)

\(\displaystyle 7\)

Correct answer:

\(\displaystyle 7\)

Explanation:

The median is the middle value of the set in increasing order.

In this set of 6 (or any even number of) entries, the median is the mean of the two middle entries of the set in increasing order

\(\displaystyle \frac{6+8}{2}\)

or

\(\displaystyle 7\)

Example Question #1 : How To Find The Median For A Set Of Data

Let \(\displaystyle b\) be a positive integer.

Find the median of the set.

\(\displaystyle b,3b,5b,7b,9b,11b,13b,15b\)

Possible Answers:

\(\displaystyle 11b\)

\(\displaystyle 10b\)

\(\displaystyle 8b\)

\(\displaystyle 9b\)

Correct answer:

\(\displaystyle 8b\)

Explanation:

The median is the middle value of the set in increasing order.

In this set of 8 (or any even number) entries, the median is the mean of the two middle entries of the set in increasing order

\(\displaystyle \frac{7b+9b}{2}\)

or

\(\displaystyle 8b\)

Example Question #1 : Univariate Data

Suppose a basketball team plays six games and scores the following points: 69, 78, 82, 69, 98, 85. Find the median.

Possible Answers:

75.5

82

80

69

Correct answer:

80

Explanation:

To find the median of a sample with an even sample size, order the values from smallest to largest, take the two values in the middle, add them, and divide that by zero. 

69, 78, 82, 69, 98, 85 

69, 69, {78, 82} 85, 98 

(78+82) / 2 = 80

Example Question #1 : Data

A sample consists of the following observations:\(\displaystyle (2, 8, 5, 7, 2)\). What is the mean? 

Possible Answers:

\(\displaystyle 8.4\)

\(\displaystyle 5\)

\(\displaystyle 4.8\)

\(\displaystyle 2.7\)

\(\displaystyle 2\)

Correct answer:

\(\displaystyle 4.8\)

Explanation:

The mean is \(\displaystyle (2+8+5+7+2)/5= 2.8.\)

Example Question #2 : Data

A bird watcher observed how many birds came to her bird feeder for four days.  These were the results:

Day 1: 15

Day 2: 12

Day 3: 10

Day 4: 13

What is the mean number of birds?

Possible Answers:

\(\displaystyle 12.5\)

\(\displaystyle 11\)

\(\displaystyle 11.5\)

\(\displaystyle 13\)

\(\displaystyle 14.5\)

Correct answer:

\(\displaystyle 12.5\)

Explanation:

The mean is calculated by adding the values from the four days together, then dividing the sum by the total number of values.  In this case, four values must be added:

\(\displaystyle 15+12+10+13=50\)

\(\displaystyle 50/4=12.5\)

Example Question #1 : Data

What is the mode of the following list of numbers?

35, 17, 4, 25, 7, 4, 17, 26, 8, 17

Possible Answers:

\(\displaystyle 31\)

\(\displaystyle 15\)

\(\displaystyle 16\)

\(\displaystyle 4\)

\(\displaystyle 17\)

Correct answer:

\(\displaystyle 17\)

Explanation:

The mode is the number that appears most frequently in a series of numbers.  First, organize the numbers in order from least to greatest: 4, 4, 7, 8, 17, 17, 17, 25, 26, 35.  The number 17 appears three times, more than any other number.

Example Question #1 : Univariate Data

Find the mean of the following set.

\(\displaystyle 1,2,3,5,7,8,9,9,9,10\)

Possible Answers:

\(\displaystyle 7\)

\(\displaystyle 5\)

\(\displaystyle 6.3\)

\(\displaystyle 7.3\)

Correct answer:

\(\displaystyle 6.3\)

Explanation:

To find the mean of a set, follow the formula

\(\displaystyle \frac{\sum_{i=1}^{n}x_i}{n}\)

in this case

\(\displaystyle \frac{1+2+3+5+7+8+9+9+9+10}{10}\)

or

\(\displaystyle 6.3\)

Example Question #2 : Data

A business tracked the number of customer calls received over a period of five days.  What was the mean number of customer calls received in a day?

Day 1: 57

Day 2: 63

Day 3: 48

Day 4: 49

Day 5: 59

Possible Answers:

\(\displaystyle 53.8\)

\(\displaystyle 276\)

\(\displaystyle 55.2\)

\(\displaystyle 57.2\)

\(\displaystyle 15\)

Correct answer:

\(\displaystyle 55.2\)

Explanation:

The mean is calculated by adding all the values in a data set, then dividing the sum by the total number in the group. 

 

\(\displaystyle 57+63+48+49+59=276\)

 

\(\displaystyle 276/5=55.2\)

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