Unit 07: Statistics

6.11 a) represent the mean of a data set graphically as the balance point; and b) determine the effect on measures of center when a single value of a data set is added, removed, or changed.

Mean, Median, and Mode

Type: mean (11,12,93,98,100,92,88,98) into DESMOS and see what happens.

What is a measure of center?

Measures of Center - types of averages for a data set. Three measures of center are the mean, median, and mode.

The Balance Point - the mean is also refered to as a balance point. Although, the mean, median, and mode are types of averages, only the mean is called the balance point.

How do I choose which measure of center to use?

Mean works best on data with no outliers, numbers that are far away from the majority of the data set. In the following data set, 11 and 12 are the outliers.

(11, 12, 93, 93, 95, 98, 100, 92, 88, 98)

Without the two outliers, the student has a 95% grade in the class; with them, the student has a 78% grade.

Median works best where there are a few very high or very low.

(11, 12, 88, 92, 93, 93, 95, 98, 98, ~~100~~)

The median would give this student a 93% grade in the class.

Mode is a good descriptor for categorical data. It is the number that occurs the most often. Our example is bimodal because 93, and 98 tie for most often.

(11, 12, 93, 93, 95, 98, 100, 92, 88, 98)

Think About It

Although, in this example, 93 and 98 with both give the student an "A" in the class, why wouldn't mode be a good thing to use for grades?

What if your outliers were both 11 and you didn't have any other matching grades?

Quantitative Data

Quantitative -vs- Categorical Data

Numerical Data (also called Quantitative Data) -vs- Categorical Data

Categorical data has no logical order, and can't be translated into a numerical value. Eye color is an example, because 'brown' is not higher or lower than 'blue'. A bar graph, for example, uses categorical data.

Quantitative or numerical data are numbers, and that way they can be ordered. Age, weight, and height are examples of numerical data. A histogram is an example of a graph that uses quantatative data.

6.10 given a practical situation, a) represent data in a circle graph; b) make observations and inferences about data represented in a circle graph; and c) compare circle graphs with the same data represented in bar graphs, pictographs, and line plots.

Circle Graphs - Part of a Whole

Interpret the Circle Graph

I DO:

What is the ratio of squash to the total plants in the garden?

That would be the number of squash divided by all the plants, including squash. You can use DESMOS to simplify your answer by clicking on "convert to a fraction" located to the right of your decimal answer.

You will be able to use DESMOS for all your statistics questions.

YOU DO:

Now you try. Open a desmos.com tab and choose the scientific calculator.

What is the ratio of tomatoes to total plants in the garden?

Where you able to simplify your answer?

*Go to the answer key at the bottom of this page to see if you got it right.

I DO:

What percent of the plants are lettuce?

Divide 100 lettuce plants by 215 total plants and round your answer to the hundredths place. Percent means out of 100, so rewrite your decimal as percent by moving the decimal two places to the right and writing a percent sign. The answer is 47%.

Helpful Trick

DECIMAL PERCENT

In the alphabet, the letter "D" comes before the letter "P". When you are turning a percent into a decimal, the decimal point moves two places to the left; "D" is to the left of "P". When you are turning a decimal into a percent, the decimal point moves two places to the right; "P" is to the right of "D".

Mathematical Thinking

Percent means out of 100. To remove a percent sign, the number must be divided by 100, moving the decimal two places to the left. To add a percent sign, the number must be multiplied by 100, moving the decimal two places to the right.

Finding Information using a Circle Graph

Percents in a Pie Graph (1).mp4

Common Mistake

Students commonly believe that all the numbers add to 100, not just the percents.

Sometimes, the actual survey numbers are given or ratios are given instead of percents. Survey numbers will add to however many people were surveyed. It could be that 20 people are surveyed or thousands. Survey amounts will not add to 100 unless 100 people were surveyed.

How to use a protractor

Finding the % of a number

Percents in a Circle Graph

All circle graphs are equal to 100% of something. Therefore, all percentages in a circle graph will always add to 100%.

How do I find the percent of a number using DESMOS?

Type the percent number, followed by the % sign. % of will appear. Then, type the total amount. DESMOS will take the % of that total.

Practice:

Type the number 15 in the DESMOS calculator below. Then click on the % key. DEMSOS will type out a % followed by the word "of". Now type the number 200 after the word "of". On the right, DESMOS should have told you what 15% of 200 is.

Note: The % sign in DESMOS DOES NOT turn a number into a percent. It is only used to find the percent of a number.

Want to do it without using the % key?

Turn the percent into a decimal and multiply by the total number of people surveyed. 0.15 x 200 = 30

Caution: DO NOT multiply by 15. 15% is 15 hundredths, NOT 15 whole.

Will you need to use a protractor on the SOL?

There are other ways to measure an angle. So, you could use one of those methods, but it is always good to know more than one way to do something.

Creating a Circle Graph

How to Make a Circle Graph https://www.mathsisfun.com/data/pie-charts.html

How to use a Circle Graph to make Predictions

Bar Graphs

What is a bar graph for? Bar graphs measure categorical data. If I wanted to create a visual representation of the number of people who died in the civil war compared to other wars, that number would stand out on a bar graph.

7.9 given data in a practical situation, a) represent data in a histogram; b) make observations and inferences about data represented in a histogram; and c) compare histograms with the same data represented in stem-and-leaf plots, line plots, and circle graphs.

Bar Graphs -vs- Histograms

What is a histogram for? Remember when we talked about quantitative data and categorical data? Bar graphs measure categorical data. Histograms measure quantitative data.

Waze

I use a phone application that brings up a histogram of traffic patterns. I look at this to decide on the best time to leave to avoid traffic.

More Differences:

Each bar in a histogram is in sections called intervals, also known as, bins, buckets, classes, and class intervals. We are going to call them intervals.

Histograms have intervals. Intervals group information. This group of numbers could be dates, time, or shoe sizes. In a histogram, these numbers are always in order. A bar graph has individual categories that could be in any order.

Bar graphs have spaces between the bars. The only time you will see a space between two histogram bars is when there is no data for that interval.

Create Graphs using the Math is fun website.

https://www.mathsisfun.com/data/data-graph.php

Are you able to find the outliers on this histogram?

They are the only two scores are between 10 and 20; all the other percentages are between 80 and 100.

Stem-and-leaf in the real world

Unit 08: Statistics

Key Vocabulary

6.10, 6.11, 7.9

bar graph - a graph in which information is displayed as rectangular bars or objects; a good choice for comparing countable populations.
circle graph (pie chart) - a graph in which information is shown as parts of a circle; most useful for comparing parts of a whole.
line plot - a vertical graph that uses columns of Xs above a number line to show data.
pictograph (picture graph) -
histogram - A graphical display where the data is grouped into intervals (such as "100 to 149", "150 to 199", etc), and then plotted as bars. The height of each bar shows how many are in each range.
frequency distribution - A table that lists a set of scores and their frequency, how many times each one occurs.
intervals - the numbers between two given numbers and may or may not include the starting and ending points. The Civil War lasted from 1861 to 1865. This interval of time includes 1861 and 1865.
stem-and-leaf plot - a tabular arrangement of numerical data that separates the data by tens (or by place value). The right-most column of the table, containing the ones digit (or least place-value digit) is called the leaf. The left-most column, containing the greater value digits is called the stem; most useful for comparing data that involve place value.
tally marks - A way of keeping count by drawing marks. Every fifth mark is drawn across the previous 4 marks, so you can easily see groups of 5. ~~IIII~~
numerical data - numerical data is measurable, such as time, height, weight, and amount.
categorical data - data that can't be translated into a numerical value. For example, chocolate milk is a category.
discrete data - data that can only take certain values. The number of students in a class is discrete because you can't have half of a student.
continuous data - data that can take any value within a range. The height of a student in class is continuous because there are many values between 5'1" and 5'2".
mean - The mean is the average of the numbers: a calculated "central" value of a set of numbers. To calculate it add up all the numbers and divide by how many numbers there are.The mean works well when data sets do not have values much higher or lower than most of the other values, no outliers.
median - The "middle" of a sorted list of numbers. To find the Median, place the numbers in value order and find the middle number. The median is a good choice when data sets have a couple of values much higher or lower than most other values, outliers.
mode - The number that occurs the most often. Mode is a good descriptor when the set of data has some identical values.
bimodal - two modes
measures of center - numbers that describe a data set. Mean, median, and mode are the measures of center that are useful for describing a typical value for different situations.
balance point - the mean is the balance point.

PRobability and Statistics (6th Grade)

Students will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to:

6.11 a) represent the mean of a data set graphically as the balance point; and b) determine the effect on measures of center when a single value of a data set is added, removed, or changed.

PRobability and Statistics (7th Grade)

SOL 6.10a,b,c Study Guide.pdf

SOL 6.11a,b Study Guide.pdf

SOL 7.9 Study Guide.pdf

Not all graphs look alike, even when they are the same type of graph.

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