Basic Statistics - Algebra 2

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Question

Determine the interquartile range of the following numbers:

42, 51, 62, 47, 38, 50, 54, 43

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Answer

How do you find the interquartile range?

We can find the interquartile range or IQR in four simple steps:

Order the data from least to greatest

Find the median

Calculate the median of both the lower and upper half of the data

The IQR is the difference between the upper and lower medians

Step 1: Order the data

In order to calculate the IQR, we need to begin by ordering the values of the data set from the least to the greatest. Likewise, in order to calculate the median, we need to arrange the numbers in ascending order (i.e. from the least to the greatest).

Let's sort an example data set with an odd number of values into ascending order.

$\textup{Odd data set}: 9, 3, 2, 5, 6, 11, 4, 3, 2$

$\textup{Odd data set (ascending)}: 2, 2, 3, 3, 4, 5, 6, 9, 11$

Now, let's perform this task with another example data set that is comprised of an even number of values.

$\textup{Even data set}: 11, 2, 4, 3, 8, 1, 2, 7, 4, 9$

Rearrange into ascending order.

$\textup{Even data set (ascending)}: 1, 2, 2, 3, 4, 4, 7, 8, 9, 11$

Step 2: Calculate the median

Next, we need to calculate the median. The median is the "center" of the data. If the data set has an odd number of data points, then the mean is the centermost number. On the other hand, if the data set has an even number of values, then we will need to take the arithmetic average of the two centermost values. We will calculate this average by adding the two numbers together and then dividing that number by two.

First, we will find the median of a set with an odd number of values. Cross out values until you find the centermost point

$\textup{Odd data set}: \not{2}, \not{2}, \not{3}, \not{3}, {\color{Red} 4}, \not{5}, \not{6}, \not{9}, \not{11}$

The median of the odd valued data set is four.

Now, let's find the mean of the data set with an even number of values. Cross out values until you find the two centermost points and then calculate the average the two values.

$\textup{Even data set}: \not{1}, \not{2}, \not{2}, \not{3}, {\color{Red} 4}, {\color{Red} 4}, \not{7}, \not{8}, \not{9}, \not{11}$

Find the average of the two centermost values.

$\textup{Average}=\frac{4+4}{2}$

$\textup{Average}=\frac{8}{2}$

$\textup{Average}=4$

The median of the even valued set is four.

Step 3: Upper and lower medians

Once we have found the median of the entire set, we can find the medians of the upper and lower portions of the data. If the data set has an odd number of values, we will omit the median or centermost value of the set. Afterwards, we will find the individual medians for the upper and lower portions of the data.

$\textup{Odd data set}: 2, 2, 3, 3, 4, 5, 6, 9, 11$

Omit the centermost value.

$\textup{Odd data set}: 2, 2, 3, 3,\mid 5, 6, 9, 11$

Find the median of the lower portion.

$\textup{Odd data set}: \not{2}, {\color{Red} 2}, {\color{Red} 3}, \not{3},\mid 5, 6, 9, 11$

Calculate the average of the two values.

$\textup{Average}=\frac{2+3}{2}$

$\textup{Average}=\frac{5}{2}$

$\textup{Average}=2.5$

The median of the lower portion is

Find the median of the upper portion.

$\textup{Odd data set}: 2, 2, 3, 3,\mid \not{5}, {\color{Red} 6}, {\color{Red} 9}, \not{11}$

Calculate the average of the two values.

$\textup{Average}=\frac{6+9}{2}$

$\textup{Average}=\frac{15}{2}$

$\textup{Average}=7.5$

The median of the upper potion is

If the data set has an even number of values, we will use the two values used to calculate the original median to divide the data set. These values are not omitted and become the largest value of the lower data set and the lowest values of the upper data set, respectively. Afterwards, we will calculate the medians of both the upper and lower portions.

$\textup{Even data set}: 1, 2, 2, 3, 4\mid 4, 7, 8, 9, 11$

Find the median of the lower portion.

$\textup{Even data set}: \not{1}, \not{2}, {\color{Red} 2}, \not{3}, \not{4}\mid 4, 7, 8, 9, 11$

The median of the lower portion is two.

Find the median of the upper portion.

$\textup{Even data set}: 1, 2, 2, 3, 4\mid \not{4}, \not{7}, {\color{Red} 8}, \not{9}, \not{11}$

The median of the upper portion is eight.

Step 4: Calculate the difference

Last, we need to calculate the difference of the upper and lower medians by subtracting the lower median from the upper median. This value equals the IQR.

Let's find the IQR of the odd data set.

$\textup{IQR of the odd data set}=7.5-2.5$

$\textup{IQR}=5$

Finally, we will find the IQR of the even data set.

$\textup{IQR of the even data set}=8-2$

$\textup{IQR}=6$

In order to better illustrate these values, their positions in a box plot have been labeled in the provided image.

Now that we have solved a few examples, let's use this knowledge to solve the given problem.

Solution:

First reorder the numbers in ascending order:

38, 42, 43, 47, 50, 51, 54, 62

Then divide the numbers into 2 groups, each containing an equal number of values:

(38, 42, 43, 47)(50, 51, 54, 62)

Q1 is the median of the group on the left, and Q3 is the median of the group on the right. Because there is an even number in each group, we'll need to find the average of the 2 middle numbers:

$Q1=\frac{42+43}{2}=\frac{85}{2}=42.5$

$Q3=\frac{51+54}{2}=\frac{105}{2}=52.5$

The interquartile range is the difference between Q3 and Q1:

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Question

Find the range of the set:

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Answer

To find the range of a set subtract the smallest number in the set from the largest number in the set:

${\color{Blue} 12}, 15, 16, 24, 25, 31, {\color{Green} 36}$

The largest number is in green:

The smallest number is in blue:

Therefore the range is,

.

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Question

Find the range of the set:

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Answer

To find the range of a set subtract the smallest number in the set from the largest number in the set:

${\color{Blue} 15}, 24, 27, 28, 28, 39, 44, 47, 61, 77, {\color{Green} 88}$

The largest number is in green:

The smallest number is in blue:

Therefore the range is,

.

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Question

Find the range of the following set of data.

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Answer

To find the range of a set subtract the smallest number in the set from the largest number in the set:

${\color{Blue} 14}, 22, 22, 31, 32, 35, 36, 37, 39, 41, 42, 48, 50, 58, {\color{Green} 63}$

The largest number is in green:

The smallest number is in blue:

Therefore the range is their difference,

.

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Question

Find the range of the set:

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Answer

To find the range of a set subtract the smallest number in the set from the largest number in the set:

${\color{Blue} 2}, 3, 5, 8, 13, 21, 34, {\color{Green} 55}$

The largest number is in green:

The smallest number is in blue:

Therefore the range is their difference,

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Question

Find the range of the set:

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Answer

To find the range of a set subtract the smallest number in the set from the largest number in the set:

${\color{Blue}15}, 15, 19, 21, 22, 24, 26, 27, 29, 31, 33, {\color{Green}36}$

The largest number is in green:

The smallest number is in blue:

Therefore the range is their difference,

.

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Question

What is the range of the following data set?

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Answer

What is the range of the following data set?

The range is found by taking the difference between the largest and smallest value in the data set.

Largest: 103

Smallest: 1

So our range is

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Question

Find the range of the following dataset:

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Answer

The range is the difference of the largest and the smallest number.

The largest number in this set is . The smallest number in this set is .

Subtract these numbers.

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Question

Find the range of the dataset.

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Answer

The range of the dataset is the difference of the highest and lowest numbers. Determine the highest number. The highest number in the set is .

The lowest number is .

Subtract these numbers.

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Question

What is the range of the set:

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Answer

Step 1: Rearrange the numbers from smallest to largest:

After rearranging, the set is: .

Step 2: Locate the highest number and the lowest number

Highest=
Lowest=

Step 3: Subtract Lowest from Highest

The range of this set of numbers is .

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Question

Find the range of the dataset: $a = [\frac{1}{4}, \frac{5}{6} , \frac{2}{3},\frac{3}{8} ]$

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Answer

The range is the difference of the highest and lowest number. In order to determine the highest and lowest fraction in the dataset, we must convert each fraction to a like denominator and compare.

The least common denominator for these fractions is . Reconvert all fractions with a denominator of 24 in order to compare numerators. Multiply the numerators with what was multiplied on the denominator to get the least common denominator.

$a = [\frac{1}{4}, \frac{5}{6} , \frac{2}{3},\frac{3}{8} ] = [\frac{6}{24}, \frac{20}{24} , \frac{16}{24},\frac{9}{24} ]$

The largest number is: $\frac{5}{6}$ or $\frac{20}{24}$

The smallest number is: $\frac{1}{4}$ or $\frac{6}{24}$

Subtract these numbers.

$\frac{20}{24}-\frac{6}{24} = \frac{14}{24} = \frac{7}{12}$

The range is: $\frac{7}{12}$

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Question

Find the range of the following data set:

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Answer

Find the range of the following data set:

The range is simply the distance between the largest and smallest value.

Let's begin by finding our two extreme values:

Largest: 2952

Smallest: 1

So our range is 2951

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Question

Find the range of this data set:

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Answer

Find the range of this data set:

To begin, let's put our numbers in increasing order:

Next, find the difference between our largest and smallest number. This is our range:

So our answer is 922

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Question

Find the median of the set:

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Answer

The median is the middle number of the set, when it is listed in order from smallest to largest or vice versa. In this case we have an even amount of numbers in the set meaning there are two "middle numbers"- 8 and 13. In order to find the median we take the average of 8 and 13:

$\frac{8+13}{2}=10.5$

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Question

Find the range of the following data set:

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Answer

Find the range of the following data set:

Let's begin by putting our data in increasing order:

Next, find the difference between our first and last numbers. This will be our range.

So our answer is 566

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Question

Find the mode of the set:

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Answer

The mode is the most repeated number in the set. To answer this, find the number that occurs the most number of times in the set. For this problem that gives us the answer of 11 for the mode.

$10, 10, {\color{Red} 11}, {\color{Red} 11}, {\color{Red} 11}, 12, 12, 15, 27, 39$

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Question

Find the mean of the set:

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Answer

To find the mean, there are two steps:

1. Add all of the numbers in the set together.

2. Divide that number by the amount of numbers are in the set.

For this problem there are 11 numbers in the set, so we add the numbers together and divide by 11 to find the mean:

$\frac{32+33+33+34+35+37+38+41+42+45+47}{11}=37.9$

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Question

Draw a Box and Whisker plot for the following data set.

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Answer

Put the data in numerical order (from smallest to largest) if it isn't already. In order to find the median, divide the data into two halves. In order to divide the values into quartiles, find the median of the two halves.

1st quartile: , ,

Median of 1st quartile:

2nd quartile = Median of total set:

3rd quartile: , ,

Median of 3rd quartile:

To construct the Box and Whisker Plot we use the minimum and the maximum value in the data set as the ends of the whiskers. To construct the box, we plot a line at the median of the 1st quartile, the median of our total data set, and at the median of the 3rd quartile. Then we connect the tops and bottom of the lines. The result is as follows:

The endpoints (black dots) represent the smallest and largest values, in this case, 2 and 39.

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Question

Draw a Box and Whisker plot for the following data set.

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Answer

Put the data in numerical order (from smallest to largest) if it isn't already. In order to find the median, divide the data into two halves. In order to divide the values into quartiles, find the median of the two halves.

1st quartile:

Median of 1st quartile: $\frac{11+17}{2}=\frac{28}{2}=14$

2nd quartile = Median:

3rd quartile:

Median of 3rd quartile: $\frac{30+31}{2}=\frac{61}{2}=30.5$

To construct the Box and Whisker Plot we use the minimum and the maximum value in the data set as the ends of the whiskers. To construct the box, we plot a line at the median of the 1st quartile, the median of our total data set, and at the median of the 3rd quartile. Then we connect the tops and bottom of the lines. The result is as follows:

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Question

Draw a box and whisker plot for the following data set.

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Answer

Put the data in numerical order (from smallest to largest) if it isn't already. In order to find the median, divide the data into two halves. In order to divide the values into quartiles, find the median of the two halves.

1st quartile:

Median of 1st quartile:

2nd quartile = Median:

3rd quartile:

Median of 3rd quartile:

To construct the Box and Whisker Plot we use the minimum and the maximum value in the data set as the ends of the whiskers. To construct the box, we plot a line at the median of the 1st quartile, the median of our total data set, and at the median of the 3rd quartile. Then we connect the tops and bottom of the lines. The result is as follows:

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