Unit # 4: Statistics 6th grade Statistics is the study of how to collect, organize, analyze, and interpret data

“We are rapidly entering a world where everything can be monitored and measured but the big problem is going to be the ability of humans to use, analyze, and make sense of the data.” The New York Times August 6, 2009

Essential Questions 1. How do we properly collect data? 2. How do we organize data? 3. How do we analyze data? 4. How do we interpret results?

To properly collect data, we must first ask a question that will result in multiple and varying answers rather than a single answer. A statistical question will result in variability in data and involve a real-world context.

Sort the following questions into two groups: statistical & non-statistical 1. How old is the oldest student in our class? 2. Are there more boys or girls in our class? 3. How old are the students in our class? 4. How long do students in this class spend on math homework each week? 5. How many students in our class like to watch scary movies? 6. What types of movies are preferred by students in our class?

Statistical or Non-Statistical 3. How old are the students in our class? 4. How long do students in this class spend on math homework each week? 6. What types of movies are preferred by students in our class? Non-Statistical 1. How old is the oldest student in our class? 2. Are there more boys or girls in our class? 5. How many students in our class like to watch scary movies?

Get into groups of 3 Decide whether the questions in your envelope are statistical or non-statistical. Discuss. If a question is non-statistical, discuss how you could rewrite it to be a statistical question that contains multiple and varying answers. http://grade6commoncoremath.wikispaces.hcpss.org/ Unit+5+Statistics+Probability 6.SP.1 Questions for Sorting

Pick one of the following topics 1. TV habits 2. Cell phone usage 3. Eating fruits and vegetables 4. Watching sports Write a statistical question and a non-statistical question about your topic.

Day 2

Bell work: Write a statistical question you could ask your classmates about the movies they like.

You will have 10 minutes to ask your classmates the statistical question you wrote during bell work. Create a line plot or a tally chart to display the results. Line plot Tally chart

Statistical questions can be divided into two groups: A statistical question is a question that will result in multiple and varying answers rather than a single answer. Statistical questions can be divided into two groups: categorical or numerical.

Categorical data Categorical data is measured qualitatively by placing the items into categories or groups. Ex. Favorite color, male or female, hair color Categorical data is often displayed using a bar graph or a circle graph

Categorical data Often times, categorical data does not involve numbers, therefore will not have a mean, median, or range. The mode is the answer that occurs most often. The mode is the most useful calculation for categorical data.

Numerical Data Numerical data is measured quantitatively and has a value or number for which operations such as addition or averaging make sense. Numerical data can be analyzed to find mean, median, range, and other forms of measurement Numerical data is often displayed using a line plot (dot plot), histogram, or box and whisker plot. *We will study each of these in more detail throughout this unit

Examples of numerical data graphs

Sort the questions from your envelope into two groups: Get into groups of 3 Sort the questions from your envelope into two groups: Numerical Categorical http://grade6commoncoremath.wikispaces.hcpss.org/Unit+5+Statistics+Probability 6.SP.1 Categorical and Numerical Questions

Identify whether the following graphs are categorical or numerical

Day 3

Activity: In five days, it snowed 4 inches, 3 inches, 5 inches, 1 inch, and 2 inches. 4 in 3 in 5 in 1 in 2 in Move the cubes until each stack has the same number of cubes. Oklahoma Math Connects Course 1 – Glencoe 2011 – pg. 102

Activity Summary: 1. On average, how many inches did it snow per day in five days? Explain your reasoning. 2. Suppose on the sixth day it snowed 9 inches. If you moved the cubes again, how many cubes would be in each stack. Oklahoma Math Connects Course 1 – Glencoe 2011 – pg. 102

Mean (or average) – the results from equally distributing all the data in a set. The mean can be found in either of the following ways The mean can be found by leveling all the values to have an equal distribution. http://www.shodor.org/i nteractivate/activities/Pl opIt/ The mean is the sum of the data divided by the number of pieces of data. Ex. Data set: 4, 3, 2, 5, 1 Mean: 4 + 3 + 2 + 5 + 1 5 = 15/5 Mean = 3

Find the mean of the data sets below. Years of experience on a baseball team Cell phone usage 478 295 780 685 570 588 495 390 587 376 4 6 3 2 1 0 6 4 5 3 2

Outliers Outliers are values that are much higher or lower than others in a data set Chapter 5 Test Scores 1. Calculate the mean of the data set. 2. Calculate the mean leaving out the score of 0. 3. Calculate the mean leaving out both the 20 and the 0. 100 99 98 96 95 88 86 81 79 76 66 64 52 20 0 Answers: 1. 73.3, 2. 78.6, 3. 83.1

Identify the outlier(s) in these costs: $115, $125, $55, $135, $400, $105, $115, $140 Calculate the mean of the data set. Calculate the mean of the data set without the outlier(s). Answers: The outliers would be $400 and $55. The mean of the entire data set is $148.75. The mean without the outliers is $122.50.

Write at least three sentences describing how outliers can distort calculations of the mean.

Day 4

Bell work Allison’s Math Test Scores The table shows Allison’s scores on four tests. What score does she need on the 5th test to have an overall mean score of 92? Test Scores 1 89 2 98 3 85 4 94 5 ?

One possible solution strategy 1. Add up the points for the first 4 test scores. 89 + 98 + 85 + 94 = 366 Find out how many points it would take to get a 92 average. 92 x 5 = 460 Subtract to find the points needed on the 5th test. 460-366 = 94

Analyzing Numerical Data A set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. CENTER - A measure of center summarizes all of the values of a data set with a single number (mean, median, mode) SPREAD - Describes how spread out or varied the data is (range) OVERALL SHAPE – Skewed left, skewed right, normal distribution, uniform distribution, bimodal distribution

Measures of Center Mean – also called the average, fair share or balance point of a set of data – it can be found using a leveling strategy or by finding the sum of the data divided by the number of pieces of data. Median – the middle number in a set of data ordered from smallest to largest. If the data set has an odd number of elements, the median is the single middle value. If the data set has an even number of elements, the median is the average of the two middle values. Mode – the number(s) that occur(s) most often (there can be more than one mode in a data set)

Measures of Spread or Variability Range – the difference between the greatest and least values of the set Mean Absolute Deviation- the absolute value of each data point from the mean of the data set

Hey diddle diddle The median’s in the middle; You add and divide for the mean. The mode is the one that appears the most And the range is the difference between.

Find the mean, median, mode & range.

Points scored by a basketball team Find the mean number of points scored. Find the median number of What do the values of the mean and median tell you about the overall shape or distribution of the data? Player Points Ethan 16 Collin 9 Nathan 10 Mason 4 Tyler 12 Aaron 8 Cole 2

Puzzle Times Each of the 20 students in Mr. Anderson’s class timed how long it took them to solve a puzzle. Their times (in minutes) are listed below: Display the data using a line plot. Find the mean and median of the data. Does it surprise you that the values of the mean and median are not equal? Explain why or why not. http://www.illustrativemathematics.org/illustrations/877 Student 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Time (in minutes

Overall Shape Bimodal Distribution Normal Distribution Uniform Distribution Skewed Right Skewed Left

Day 5 2nd day on previous material Also, we need to address when finding the mean is the best strategy, when finding the median is best, and when finding the mode is best.

This measure is most useful when: Mean – the data have no outliers (values much larger or smaller than the rest of the data) Median – the data contains outliers Mode – data have many repeated numbers or the data is categorical in nature

Which is the best measure of center for this data set?

Which is the best measure of center for this data set?

Which is the best measure of center for this data set?

Glencoe Pre-Algebra (2012) pg. 779 1. Jamal said that the number that best represented the following set of data is 27. Which measure of central tendency is he referring to? 28, 32, 21, 25, 33, 32, 20, 26 2. The number of books read by the students in each core literacy class is: 104, 90, 162, 134, 110, 97, 145, 126. Which measure of central tendency best describes the data? Explain. Glencoe Pre-Algebra (2012) pg. 779

3. The high temperatures for one week are 79°, 81°, 77°, 81°, 82°, 75°, and 76°. If the temperature on the eighth day is 80°, which of the following would be true? A. The mode will change B. The mean will increase and the median will remain the same C. The median will increase and the mean will remain the same D. Both the mean and the median will increase Glencoe Pre-Algebra (2012) pg. 779

Day 6

Bell work: *ACT Aspire Sample Question Nigel’s class placed 10 empty rain gauges on the playground Monday morning. The line plot below shows the number of inches of rainwater in each gauge after it rained Monday afternoon. Number of Inches of Rainwater x x x x x x x x x x 3/8 1/2 5/8 3/4 7/8 What is the mean amount of rainwater per gauge, in inches, in the 10 rain gauges?

What is the mean amount of rainwater per gauge, in inches, in the 10 rain gauges? B. 5/8 C. 51/80 D. 37/56 E. 51/8 Answer: C

Mean absolute deviation Measures the average amount that the items of a data set differ from the mean of the set. Find the mean Find the difference of each value from the mean Subtract the smaller value from the larger value, so each difference is positive Find the mean absolute deviation by averaging the differences

Hours worked on a Saturday: 4, 5, 7, 12 Mean = (4 + 5 + 7 + 12)/ 4 = 7 Deviations from the mean = 3, 2 , 0, 5 Mean absolute deviation = (3 + 2 + 0 + 5)/4 = 10/4 = 2.5

Day 7 and 8

Bar Graph See page 81

Bar graphs can be used to display categorical or numerical data.

The vertical axis represents the frequency or number of observations Histograms A histogram is a special type of bar graph used to display numerical data that has been organized into intervals. The heights of the bars show the number of people in each group. The horizontal axis represents the intervals. The vertical axis represents the frequency or number of observations

To construct a histogram 1. Draw and label a horizontal and vertical axis. Use equal intervals Every interval from the lowest value to highest value must be included even if it has a frequency of 0. 2. Include a title 3. For each interval, draw a bar whose height is given by the frequencies. There is no space between the bars

Ages of people who entered a store Frequency 1-10 5 11-20 8 21-30 14 31-40 18 41-50 20 51-60 13 61-70 6 Display the data in a histogram

Days 9 and 10

Pre-View of a box and whisker plot

Pre-View of a box and whisker plot What is the scale and interval of the number line below the plot? Are you able to tell how many parts the data set is divided into? What do you think the far right and far left points represent (at the end of the whiskers)? What do you think the line inside the box represents?

Pre-View of a box and whisker plot Where do you think the following labels should go on our box and whisker plot? Upper extreme, Lower extreme, second (lower) quartile, third (upper) quartile, interquartile range, and median.

A Box and Whisker plot Uses a number line to show the distribution of a set of data. It divides a set of data into four parts using the median and quartiles. A box is drawn around the quartile values, and whiskers extend from each quartile to the minimum and maximum values that are not outliers.

Key Features of a box plot Quartiles The median is the middle quartile The median of the lower half of the data is the lower quartile. The median of the upper half of the data is the upper quartile The lower extreme forms the lower whisker. The upper extreme forms the upper whisker. Interquartile Range (IQR) is the difference between the upper quartile and the lower quartile

Outliers – data more than 1 Outliers – data more than 1.5 times the value of the interquartile range beyond the quartiles Step 1: Find the median, lower quartile, and upper quartile. Step 2: Find the interquartile range. Step 3: Multiply the interquartile range by 1.5. Step 4: Subtract the value from the lower quartile and add the value to the

Find any outliers in the data set. Animal Speeds Glencoe Pre-Algebra (2012) pg. 793 Step 1: median - 30 lower quartile - 15 upper quartile - 35 Step 2: interquartile range 35 – 15 = 20 Step 3: IQR x 1.5 20 x 1.5 = 30 Step 4: lower quartile – 30 15 – 30 = -15 upper quartile + 30 35 + 30 = 65 70 is an outlier Animal Speed (mph) Squirrel 12 Turkey 15 Elephant 25 Cat 30 Reindeer 32 Rabbit 35 Cheetah 70

Steps to creating a box plot 1. Organize the data from smallest to largest. 2. Identify the lower extreme, lower quartile, median, upper quartile, upper extreme. 3. Find the interquartile range. 4. Identify any outliers, use an asterisk(*) to indicate an outlier. It is not connected to a whisker. 5. Draw the number line, mark the scale, and label the scale. 6. Draw the box plot. 7. Give the box plot a title.

Pre-View of a box and whisker plot Write 3-5 sentences summarizing the data displayed.

Key Ideas The shortest hair measured was 9 cm. The lower quartile is 13 cm. The median is 25 cm. The upper quartile is 33 cm. The longest hair measured was 42 cm. The range is 42 cm – 9 cm or 33 cm. The interquartile range is 33 cm -13 cm = 20 cm The middle half of the data lies between 33 and 13. So the middle half of the girls measured had hair between 33cm and 13 cm. This is more variability in the lower half of the data indicating that more girls had hair lower than 25 cm than had hair longer than 25 cm.

Limitations of a box and whisker plot The box plot provides a summary of the data. It does not show the number of observations (so you can’t find the mean) nor does it indicate if a particular value was especially common (so you can’t find the mode).

Higher –Order Thinking Skills 1. The range is always, sometimes, or never affected by outliers. Justify your reasoning. 2. True or false. The interquartile range is affected by outliers of the data set. Explain your reasoning. 3. It is always, sometimes, or never possible for the mean, median, and mode to be equal? Justify your reasoning. 4. Can a data set have more than one median? Explain. Glencoe Pre-Algebra (2012) pg. 778 and 796