Single Measurement Variable (Representing & Summarizing Numerical Data Sets) Day 1

Standards: • MGSE9-12.S.ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots). Choose appropriate graphs to be consistent with numerical data: dot plots, histograms, and box plots. • MGSE9-12.S.ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, mean absolute deviation, standard deviation) of two or more different data sets. • MGSE9-12.S.ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Students will examine graphical representations to determine if data are symmetric, skewed left, or skewed right and how the shape of the data affects descriptive statistics.

Learning Target(s): 1. I can calculate mean, median, and mode. 2. I can create and interpret box, histogram, and dot plots. 3. I can interpret the shape of data plots. 4. I can interpret and compare two data sets.

Essential Questions 1. What are the different ways to represent data? 2. What are the advantages and disadvantages of various measures of central tendency? 3. How do I know the best way to represent data?

Measures of Central Tendency • A measure of central tendency is a measure that tells us where the middle of a bunch of data lies. • The three most common measures of central tendency are the mean, the median, and the mode.

• Also referred to as “Center”

Mean The average value of a data set, found by summing all values and dividing by the number of data points

Median • Median is the number present in the middle when the numbers in a set of data are arranged in ascending or descending order. If the number of numbers in a data set is even, then the median is the mean of the two middle numbers.

Median The middle-most value of a data set; 50% of the data is less than this value, and 50% is greater than it Example:

Mode • Mode is the value that occurs most frequently in a set of data.

Range • The range is the difference between the highest and lowest scores in a data set and is the simplest measure of spread. So we calculate range as: Range = maximum value - minimum value

Visualize Vocabulary

First Quartile (Q1) • The ‘middle value’ in the lower half of the data • a.k.a the lower quartile

Third Quartile (Q3) • The ‘middle value’ in the upper half of the data • a.k.a the upper quartile

Box Plot A plot showing the minimum, maximum, first quartile, median, and third quartile of a data set; the middle 50% of the data is indicated by a box. Example: *Be sure to use equal intervals (count by 1, 2, or 4 be consistent)

Box Plot: Pros &Cons Advantages: • Shows 5-point summary and outliers • Easily compares two or more data sets • Handles extremely large data sets easily Disadvantages: • Not as visually appealing as other graphs • Exact values not retained

Put the vocabulary word in the correct position. Minimum Lower quartile (Q1) Median Maximum Upper quartile (Q3)

Dot Plot A frequency plot that shows the number of times a response occurred in a data set, where each data value is represented by a dot. Example:

Dot Plot: Pros &Cons Advantages: • Simple to make • Shows each individual data point Disadvantages: • Can be time consuming with lots of data points to make • Have to count to get exact total. Fractions of units are hard to display.

Histogram A frequency plot that shows the number of times a response or range of responses occurred in a data set. Example:

Histogram: Pros & Cons Advantages: • Visually strong • Good for determining the shape of the data Disadvantages: • Cannot read exact values because data is grouped into categories • More difficult to compare two data sets

Outlier A data value that is much greater than or much less than the rest of the data in a data set; mathematically, any data less than or greater than Q1 1.5( IQR) is an outlier Example: Q3  1.5( IQR)

Interquartile Range • The interquartile range describes the difference between the third quartile (Q3) and the first quartile (Q1), telling us about the range of the middle half of the scores in the distribution.

• IQR = Q3 – Q1

Interquartile Range Example

For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120} Find the IQR (interquartile range).

• Step 1: split the data in half

• Step 2: find Q1 • Step 3: find Q3 • Step 4: Solve

IQR = Q3-Q1=15 – 6 = 9

Box & Whisker Plot The numbers below represent the number of homeruns hit by players of the Wheeler baseball team.

2, 3, 5, 7, 8, 10, 14, 18, 19, 21, 25, 28 Q1 = (5+7)/2=6

Q3 = (19+21)/2=20

Interquartile Range: 20 – 6 = 14

Box & Whisker Plot Example 4, 17, 7, 14, 18, 12, 3, 16, 10, 4, 4, 11 1. Put the numbers in order: 3, 4, 4, 4, 7, 10, 11, 12, 14, 16, 17, 18

2. Cut in half: 3, 4, 4 , 4, 7, 10 11, 12, 14 , 16, 17, 18

3, 4, 4 , 4, 7, 10 11, 12, 14 , 16, 17, 18 3. Quartile 1 (Q1) = (4+4)/2 = 4

4. Quartile 3 (Q3) = (14+16)/2 = 15 Also: • The minimum (lowest value) is 3 • The maximum (highest value) is 18

Teacher Guided Practice

Classwork Class A Test Scores: 51, 45, 45, 45, 33, 51, 48, 36, 48, 51, 27, 51, 36, 48, 51 Class B Test Scores: 48, 51, 48, 24, 48, 51, 48, 48, 51, 18, 48, 51, 48, 45, 21

Classwork

Homework

Homework

Single Measure Variable (Mean Absolute Deviation) Day 2

Mean Absolute Deviation (MAD) MAD is the distance each data value is from the mean of the data set. • If a data set has a small mean absolute deviation, then this means that the data values are relatively close to the mean. • If the mean absolute deviation is large, then the values are spread out and far from the mean.

To find the MAD: 1. Find the mean (𝐱) 2. Subtract the mean from each data value (x  𝐱) 3. Take the absolute value of each value from step #2. ( 𝐱 − 𝐱 ) 4. Add up all values from step #3. 5. Divide by the number of data values.

Classwork

Single Measurement Variable (Interpreting Data Sets) Day 3

Shape

(Negative value)

(Positive value)

Unusual Features

Classwork

4. Essential question: What are the different ways to represent data? Write 3 complete sentences in the summary section of the first page of your notes.

“Ticket out the Door”

One-Variable Statistics (Compare Data Sets) Day 4