quartiles and box plots common core algebra 1 homework answers

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Statistics and probability

Course: statistics and probability   >   unit 3.

• Worked example: Creating a box plot (odd number of data points)
• Worked example: Creating a box plot (even number of data points)
• Constructing a box plot
• Creating box plots
• Reading box plots
• Interpreting box plots

Interpreting quartiles

Box plot review.

• Judging outliers in a dataset
• Identifying outliers
• Identifying outliers with the 1.5xIQR rule

What is a box and whisker plot?

Example: finding the five-number summary, example (continued): making a box plot, example: interpreting quartiles, want to join the conversation.

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Box Plot Worksheets

Welcome to our Box Plot Worksheets page. Here you will find our range of free statistics worksheets, which will help you to learn how to create and interpret box plots.

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• What is a Box Plot?
• Creating Box Plots Worksheets
• Interpreting Box Plots Worksheets

Box Plots Online Quiz

• More recommended resources

A box plot is a visual way of recording data values.

The data from a box plot can show us 5 facts:

• the minimum data value;
• the 1st quartile value;
• the median value.
• the 3rd quartile value;
• the maximum data value;

Using these facts, we can also quickly use the box plot work out:

• the range of the data set by subtracting the minimum value from the maximum value;
• the interquartile range of the data set by subtracting the 1st quartile value from the 3rd quartile value;
• whether the data is skewed to the left or right.

If you would like some support learning about box plots, use the link below.

• What is a Box Plot? Support Page

Box Plot Worksheets Information

These sheets have been designed for students around 6th grade level.

We have split our worksheets into two sections:

• creating box plot worksheets;
• interpreting box plot worksheets.

Creating Box Plot Worksheet s

These sheets involve the following:

• putting a data set in order;
• finding the median value;
• finding the 1st (lower) and 3rd (upper) quartiles;
• finding the range and the interquartile range;
• using the information to draw a box plot on the number line provided.
• Creating Box Plots Sheet 1
• PDF version
• Creating Box Plots Sheet 2
• Creating Box Plots Sheet 3

Interpreting Box Plot Worksheets

These sheets involve answering a range of questions about the box plot, including:

• identifying the minimum and maximum values;
• identifying the median value.
• finding the range and interquartile range;
• understanding what the lower and upper quartiles represent.
• Interpreting Box Plot Worksheets 1
• Interpreting Box Plot Worksheets 2
• Interpreting Box Plot Worksheets 3

Interpreting Box Plots Walkthrough Video

This short video walkthrough shows our Interpreting Box Plots Worksheet 3 being solved and has been produced by the West Explains Best math channel.

If you would like some support in solving the problems on these sheets, please check out the video below!

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

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The box plot support page below will help you to learn all about box plots and how they work.

Lower Quartile and Upper Quartile Support

If you want some more help identifying the lower and upper quartiles, then take a look at this page.

• Lower Quartile and Upper Quartile Support Page
• Median Worksheets

Find links to our Median worksheets below.

Using this webapge will help you to:

• find the median of a set of data;
• find the median of both odd and even numbers of data points;
• show you worked examples of how to find the median.
• Mean Worksheets

The sheets in this section will help you to find the mean of a range of numbers, including negative numbers and decimals.

There are a range of sheets involving finding the mean, and also finding a missing data point when the mean is given.

• Mode and Range Worksheets

The sheets in this section will help you to find the mode and range of a set of numbers, including negative numbers and decimals.

There are easier sheets involving fewer data points, and harder ones with more data points.

• Mean Median Mode and Range Worksheets

The sheets in this section will help you to find the mean, median, mode and range of a set of numbers, including negative numbers and decimals.

• Dot Plot Worksheets

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Box and Whisker Plot Worksheets

• Box Plots on a Numberline
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Here you will learn about a quartile, including what a quartile is, how to find the lower quartile and upper quartile for a set of data, and why these measures are useful.

Students will first learn about quartiles as part of statistics and probability in 6 th grade.

Every week, we teach lessons on quartiles to students in schools and districts across the US as part of our online one-on-one math tutoring programs. On this page we’ve broken down everything we’ve learnt about teaching this topic effectively.

What is a quartile?

A quartile divides an ordered data set into four equal parts (quarters). You can use abbreviations to label the quartiles: Q1, M or Q2, and Q3.

The first quartile, Q1, is \cfrac{1}{4} \: ( or 25 \%) of the way through the data – the lower quartile .

The second quartile, M or Q2, is \cfrac{1}{2} \: ( or 50 \%) of the way through the data – the median .

The third quartile, Q3 is \cfrac{3}{4} \: ( or 75 \%) of the way through the data – the upper quartile .

You can only find quartiles for quantitative (numerical) data sets, but they can be found for both discrete and continuous quantitative data sets.

Remember, discrete data is data that can only take certain values (counted data), whereas continuous data can take any value within a given range (measured data).

For a small data set, you can find quartiles by simply counting and finding the correct data points.

For example,

[FREE] Quartile Worksheet (Grade 6)

Use this worksheet to check your grade 6 students’ understanding of quartile. 15 questions with answers to identify areas of strength and support!

• Interquartile range

Finding the lower quartile and upper quartile for a data set allows you to examine the middle half of the data centralized around the median – this is useful if a data set contains a lot of outliers or extreme values.

The lower quartile and upper quartile can also be used to calculate the interquartile range (IQR), which is a measure of the variability or spread of the data.

I Q R=U Q-L Q or I Q R=Q 3-Q 1

The median and lower and upper quartiles, along with the minimum value and the maximum value of the data set, form a five-number summary of descriptive statistics for the data set.

This information can then be presented in a box plot (also known as a box and whisker plot or diagram), making it easy to compare with other sets of data.

Percentiles

Quartiles are one way of splitting data to analyze; you may see the word percentiles used when discussing data sets in different contexts, such as news reports or test scores.

For example, you might see something about ‘the top 10 \% of test scores’ – to calculate this, you would find the 90 th percentile using a similar method to that for quartiles.

There is a link between quartiles and percentiles, as follows:

• First/lower quartile (Q1) - 25 th percentile
• Second quartile/median (M or Q2) - 50 th percentile
• Third/upper quartile (Q3) - 75 th percentile

You will use percentiles as you progress through your statistics and probability units, along with standard deviation (a measure of dispersion or central tendency), determining outliers, using the normal distribution, including quantiles and deciles, and more exploratory data analysis techniques.

Common Core State Standards

How does this relate to 6 th grade math?

• Grade 6 – Statistics and Probability (6.SP.B.4) Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

How to find a quartile for a small data set

In order to find the quartiles for a small data set:

Order the data and find the median \textbf{(Q2).}

Count the number of data items in the set. Highlight the median and find the halfway point in the lower half of the data \textbf{(Q1).}

Highlight the median and find the halfway point in the upper half of the data \textbf{(Q3).}

Quartile examples

Example 1: small discrete data set, odd number of data points.

Find the lower and upper quartiles.

The data is ordered, so you can go straight ahead and find the median. For a small data set, you can just cross numbers off from either end one by one until you reach the middle.

The median (Q2) is 7.

2 Count the number of data items in the set. Highlight the median and find the halfway point in the lower half of the data \textbf{(Q1).}

There are 11 items of data. As there is an odd number of data items, you do not include the median when looking at the lower half of the data.

If you mark 7 (the median) using two vertical lines either side of the number, as if to fence off the value from further calculations, the lower half of the data is:

Repeating the process of crossing off to find the middle value for the lower half of the data, you have:

The lower quartile (Q1) is 2.

3 Highlight the median and find the halfway point in the upper half of the data \textbf{(Q3).}

Excluding the median, the upper half of the data is:

Repeating the process of crossing off to find the middle value for the upper half of the data, you have:

The upper quartile (Q3) is 10.

Example 2: small continuous data set, even number of data points

The data below shows birth weights of 10 babies in kilograms.

Ordering the data, you get:

Now, find the median by crossing numbers off from either end one by one until you reach the middle.

There are two middle values, so you need to calculate the midpoint of these.

(3.3+3.5)\div{2}=3.4

The median (Q2) is 3.4{~kg}.

The median was 3.4. Marking this on our set of data, you still do not include this value when calculating the lower quartile, but you do find the middle of all of the values below the median.

You mark 3.4, which is the median. The lower half of the data is:

You do include the 3.3 value you used in the median calculation, as this wasn’t actually the median value.

Repeating the process of crossing off values to find the middle value for the lower half of the data, you have:

The lower quartile (Q1) is 3.0{~kg}.

Again, note that you do include the data point 3.5.

Repeat the process of crossing off to find the middle value for the upper half of the data.

The upper quartile (Q3) is 3.7{~kg}.

Example 3: small discrete data set, using midpoints for quartiles

Find the lower and upper quartiles for the following data set.

Ordering this data set from smallest to largest, you have:

As this is a small data set, just cross numbers off from either end in turn until you reach the middle.

The median (Q2) is 14.

As there is an odd number of data values, you do not use the median value when finding the other two quartiles. You mark 14, which is the median and so the lower half of the data is:

As there are two data points, 5 and 8, you need to calculate the midpoint of these two values.

\cfrac{5+8}{2}=\cfrac{13}{2}=6.5

The lower quartile (Q1) is therefore 6.5.

The values in the upper half of the data are:

Again, note that you don’t include the median data point in the upper half.

Again, there are two data points, 20 and 34, in the middle of the data, so you find the midpoint of these, which is 27.

If you weren’t sure on the midpoint, you could do the calculation:

\cfrac{20+34}{2}=\cfrac{54}{2}=27

The upper quartile (Q3) is 27.

Working with a large data set

For a large data set, crossing numbers off a list can be time-consuming and a bit confusing, particularly if the data spans over two or more lines when listed. You can use an alternative method to find the lower and upper quartiles.

For a data set containing n values:

• The lower quartile , Q1, is located at the position \cfrac{n+1}{4}.
• The median , M or Q2, is located at the position \cfrac{n+1}{2}.
• The upper quartile , Q3, is located at the position 3\times\cfrac{n+1}{4}.

Note that you are finding out the position of the lower or upper quartile. You still need to count through the data set to find which values these are .

In situations where data is grouped, this method can also be used to find the class intervals in which the lower and upper quartiles lie. This is particularly true when estimating the quartiles in a histogram.

Note: The quartile function in Google Sheets or Microsoft Excel can help determine quartiles for very large sets of data.

How to find quartiles for a large data set

In order to find the lower quartile and upper quartile for a large data set:

Order the data and find the value of \textbf{n} (the number of data points).

Use the formula \bf{\cfrac{\textbf{n}+1}{4}} to calculate the position of \textbf{Q1} and state/calculate the data value at this position.

Use the formula \bf{3\times\cfrac{\textbf{n}+1}{4}} to calculate the position of \textbf{Q3} and state/calculate the data value at this position.

Quartile for a large data set examples

Example 4: large data set.

A set of data has been arranged into a stem and leaf plot. Calculate the value of each quartile for the following data.

The data is already ordered. There are 24 data points, so n=24.

Using the formula \cfrac{n+1}{4} with n=24, you have

\cfrac{24+1}{4}=\cfrac{25}{4}=6.25

So the lower quartile lies between the 6 th and 7 th data point.

The values are 5 and 6, and the midpoint of these is 5.5, so the lower quartile is 5.5.

You use the formula 3\times\cfrac{n+1}{4} with n=24 to give

3 \times \cfrac{24+1}{4}=3 \times \cfrac{25}{4}=3 \times 6.25=18.75

Note that if you’ve already found that the lower quartile is the 6.25 th value, you can simply multiply 6.25 by 3 to get 18.75.

So the upper quartile lies between the 18 th and 19 th data points, and you use the midpoint of these values.

The values are 19 and 20, and the midpoint of these is 19.5, so the upper quartile is 19.5.

Example 5: grouped data

The heights (h) of 40 students are given in the table below. In which class intervals do the lower and upper quartiles lie?

You don’t need to order the data. There are 40 data points, so n=40.

Note, this is given information in the actual question.

Using the formula \cfrac{n+1}{4} with n=40, you have

\cfrac{40+1}{4}=\cfrac{41}{4}=10.25

So the lower quartile value lies between the 10 th and 11 th data points.

The 10 th and the 11 th values both lie in the class interval 1.5 < h \leq {1.6} because, by using the cumulative frequency, you can see that 15 items of data have a height h\leq{1.6}\text{m}.

The lower quartile lies in the group 1.5<h\leq{1.6}.

Using the formula 3\times\cfrac{n+1}{4} with n=40, you have

3 \times \cfrac{40+1}{4}=3 \times \cfrac{41}{4}=3 \times 10.25=30.75

So the upper quartile value lies between the 30 th and 31 st data points.

The 30 th and the 31 st values both lie in the class interval 1.7 < h \leq {1.8} because, by using the cumulative frequency, you can see that 28 items of data have a height h\leq{1.7}\text{m}.

The upper quartile lies in the group 1.7<h\leq{1.8}.

Example 6: quartiles from a cumulative frequency diagram

The cumulative frequency diagram below shows the distribution of building heights in a city, in meters.

Use the diagram to calculate the values for Q1 and Q3 for this set of data.

The data within a cumulative frequency diagram is already ordered from smallest to largest. The value n is the highest value for the cumulative frequency on the graph. Here, n=80.

Using \cfrac{n+1}{4} with n=80, the position of Q1 is

\cfrac{80+1}{4}=\cfrac{81}{4}=20.25

This is the value on the cumulative frequency axis. Drawing a horizontal line to the curve, and then down to the x -axis, you can see that the value for the lower quartile is approximately 30{~m} (see diagram below).

The upper quartile is located at 3\times\cfrac{n+1}{4}. When n=20

Q_{1}=3\times\cfrac{20+1}{4}=3\times\cfrac{21}{4}=3\times{20.25}=60.75

Drawing a horizontal line from the cumulative frequency axis at the value 60.25, across to the curve and then down to the x -axis, the upper quartile is approximately 42{~m} (see diagram below).

Teaching tips for quartile

• Use visual aids such as number lines, bar graphs, or dot plots to illustrate quartiles. Show students how to locate Q1, Q2, and Q3 for any given set of values. Encourage them to interpret and analyze the data based on the quartiles.
• Make the concept relatable by using real-life examples that students can connect with. For instance, use numeric values related to their favorite sports, hobbies, or interests. This helps students see the relevance of quartiles in analyzing and understanding data.
• Provide ample opportunities for students to practice finding quartiles using different data sets. Offer exercises with varying levels of difficulty to challenge and reinforce their understanding.
• Utilize interactive tools or online resources that allow for manipulating, exploring, and even calculating quartiles to manipulate and explore quartiles. For example, use virtual manipulatives or educational websites that provide interactive activities related to quartiles.

Easy mistakes to make

• Forgetting to order the data set before finding the median or quartiles The list must be in order before you start finding the key values!
• Stating the quartile value as the number given by the formula, rather than counting and finding that value in the data For example, for the data set 3, \, 4, \, 6, \, 7, \, 10, the formula for the lower quartile gives \cfrac{5+1}{4}=\cfrac{6}{4}=1.5. This tells you which data values to select (in this case, the midpoint of the 1 st and 2 nd values); do not just write Q1=1.5.

Related representing data lessons

• Representing data
• Stem and leaf plot
• Scatterplot
• Two way tables
• Time series graph

Practice quartile questions

1) Find the values of the lower and upper quartiles for the following 11 items of data.

The median is 17, so mark this, then find the median of the lower half of the data (4, \, 6, \, 6, \, 7, \, 12) and the median of the upper half of the data (18, \, 20, \, 21, \, 32,3 \, 5).

Q1 is 6 and Q2 is 21.

2) Find the lower and upper quartiles.

The median is the middle number in an ordered list. The list is already ordered, but there are two values in the middle of the data.

As they are both 13, the median is 13.

The middle of the lower half of the data (2, \, 4, \, 7, \, 10, \, 13) is 7 so the lower quartile is 7.

The middle of the upper half of the data (13, \, 15, \, 19, \, 22, \, 32) is 19 so the upper quartile is 19.

3) Find the lower and upper quartiles for the following set of data.

Ordering the data first, you have

The median is the midpoint of 0 and 4; mark this, then find the median of the lower half of the data (-3, \, -2, \, -2, \, 0) and the median of the upper half of the data (4, \, 5, \, 10, \, 19).

There are an even number of data points, so use the midpoint of the 2 nd and 3 rd for Q1 and the midpoint of the 6 th and 7 th for Q3.

4) This table shows the shoe sizes of 15 girls. Find the lower and upper quartiles.

Use the formula to find out which values to pick for the lower quartile and the upper quartile. There are 15 pieces of data, so n=15.

Q1=\cfrac{15+1}{4}=4 so the lower quartile is the 4 th data point, which is 5.

Q3=3\times\cfrac{15+1}{4}=12 so the upper quartile is the 12 th data point, which is 7.

5) Raheem writes down how long his bus journey to school takes on 20 different days. The times are given to the nearest minute. Find the lower and upper quartiles for his data set.

Use the formula to find out which values to pick for the lower quartile and the upper quartile. There are 20 data points, so n=20.

Q1=\cfrac{20+1}{4}=5.25 so by finding the midpoint of the 5 th and 6 th data points, which are both 11 (remember to use the key), the lower quartile is 11.

Q3=3\times\cfrac{15+1}{4}=15.75 so the midpoint of the 15 th and 16 th data points, which are both 18 (using the key), gives us the upper quartile of 18.

6) The weights (w) of 30 teenagers are given in the table below. Which class interval contains the upper quartile?

Using the formula 3\times\cfrac{n+1}{4} to determine the upper quartile with n=1+8+10+7+4=30, you have

Q3=3\times\cfrac{30+1}{4}=3\times\cfrac{31}{4}=23.25

So the upper quartile is between the 23 rd and 24 th data points.

Counting the cumulative frequency for the data set, you have

The 23 rd and 24 th data points both lie in the class interval 60<w\leq{70}.

Quartile FAQs

A quartile is a type of quantile that divides an ordered data set into four equal parts (quarters).

To find each quartile of a data set, first sort the data set in ascending order from smallest to largest and find the median (Q2). Find the lower quartile (Q1) by locating the median (central value) of the lower half of the data set, and find the upper quartile (Q3) by locating the median (central value) of the upper half of the data set.

The next lessons are

• Frequency table
• Frequency graph
• Sampling methods

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Quartiles And Box Plots Common Core Algebra 1 Homework

Quartiles and box plots common core algebra 1 homework – In the realm of mathematics, quartiles and box plots emerge as indispensable tools for exploring data distribution and gaining insights into the underlying patterns. This comprehensive guide delves into these concepts, providing a thorough understanding of their significance in Common Core Algebra 1 homework.

Quartiles, pivotal measures in data analysis, partition a dataset into four equal parts, unveiling the distribution’s central tendencies and spread. Box plots, on the other hand, offer a graphical representation of these quartiles, allowing for visual exploration and comparison of data sets.

Quartiles and Box Plots: Common Core Algebra 1 Homework: Quartiles And Box Plots Common Core Algebra 1 Homework

In this article, we will explore the concepts of quartiles and box plots, their significance in data analysis, and their relevance to Common Core Algebra 1 curriculum. We will also provide a comprehensive homework assignment to reinforce these concepts.

Quartiles divide a dataset into four equal parts, providing valuable insights into the distribution of data. The first quartile (Q1) represents the 25th percentile, the second quartile (Q2) represents the median, and the third quartile (Q3) represents the 75th percentile.

To calculate quartiles, we sort the data in ascending order and then use the following formulas:

• Q1 = (n+1)/4
• Q2 = (n+1)/2
• Q3 = 3(n+1)/4

where n is the number of data points.

Box plots are graphical representations of data distribution that visually display the quartiles and other key statistics. The components of a box plot include:

• Minimum: The lowest value in the dataset
• First quartile (Q1)
• Median (Q2)
• Third quartile (Q3)
• Maximum: The highest value in the dataset

To construct a box plot, we draw a horizontal line for the median and then extend lines to the quartiles. The whiskers extend from the quartiles to the minimum and maximum values.

Common Core Algebra 1

The Common Core State Standards for Mathematics (CCSSM) include learning objectives related to quartiles and box plots in Algebra 1. These objectives emphasize:

• Understanding the concept of quartiles and their use in describing data distribution
• Calculating quartiles from a given dataset
• Constructing and interpreting box plots
• Analyzing data using quartiles and box plots

Incorporating these concepts into Algebra 1 curriculum is essential for developing students’ statistical reasoning and data analysis skills.

Homework Assignment, Quartiles and box plots common core algebra 1 homework

The following homework assignment is designed to assess students’ understanding of quartiles, box plots, and their applications in data analysis.

• Calculate the quartiles of the following dataset: 5, 7, 10, 12, 15, 18, 20
• Construct a box plot for the following dataset: 4, 6, 8, 10, 12, 14, 16, 18, 20
• Explain the significance of quartiles in analyzing the distribution of a dataset
• Discuss the advantages and limitations of using box plots for data visualization
• Use quartiles and box plots to analyze the distribution of test scores in a class
• Create a data visualization that compares the distribution of two different datasets using quartiles and box plots

FAQ Explained

What is the significance of quartiles in data analysis?

Quartiles provide valuable insights into the central tendencies and spread of a dataset, aiding in identifying outliers and understanding the distribution’s shape.

How do box plots complement quartiles in data exploration?

Box plots offer a visual representation of quartiles, enabling quick comparisons between data sets and highlighting potential differences or similarities in their distributions.

Why are quartiles and box plots essential in Common Core Algebra 1?

These concepts equip students with fundamental analytical skills, fostering their ability to analyze data, make inferences, and solve real-world problems involving data interpretation.