Summarize Numerical Data Sets

Summarizing Data

Lesson Goals

• Relate the measure of center or variablility to the shape of the data distribution.

• Summarize a numerical data set, how it was measured and units of measure.

• Summarize numerical data sets by reporting the number of observations.

• Analyze data represented on a histogram.

  Interpret data on a line plot.

  Determine the interquartile range of a numerical data set.

Standards

• 6.DS.4 Summarize numerical data sets in relation to their context in multiple ways

• 6.DS.2 Select, create, and interpret graphical representations of numerical data, including line plots, histograms, and box plots.

Unit 6 Test Review!

• There are 2 parts to the Test!

• Make sure to do them both!

Multiple Choice

Is the mean to the left of the median in this chart?

• Yes

• No

Multiple Choice

What is the MEDIAN of this data

• 95

• 90

• 100

• 40

Choosing a Measure of Center

• The mean and median are both measures of center.

• When a distribution is normal or uniform the mean and median are equal (+/-) and represent the data accurately.

• If the data is skewed left or skewed right the mean is skewed away from the center, so the median is a better measure of center.

Mean or median?

• Which measure of center best represents the middle of the set of data?

• This histogram is very close to symmetrical (normal, regular)

• The mean and median are close enough that either will work as a representation of the middle of the data set.

Mean or Median?

• What is the type of distribution? The data is stretched to the right (right tailed) so it is skewed right.

• Compare the mean and median. When a distribution is skewed right the mean is greater than the median.

• Because the mean is pulled to the right the median is a better measure of center.

Mean or Median?

• What type of distribution?

• Though the heights are different neither end seems much taller or shorter. This is a uniform distribution

• It is fair to assume that the mean and median are approximately equal.

Multiple Choice

What is the shape of the data?

• symmetrical

• skewed left

• skewed right

• bell curve

Multiple Choice

How would you describe this distribution?

• Left skewed

• Uniform

• Right Skewed

• Symmetrical

Multiple Choice

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

• mean

• median

• either

Multiple Choice

What is the best representation of the center of the data

• mean

• median

• Either

Multiple Choice

What is the best representation of the center of the data

• mean

• median

• Either

Data Values

• Each x is a single data value

• The number the x is above is the value of that x.

5 Number Summary

• The 5 number summary includes the minimum, Q1, the median (Q2), Q3, and the maximum.

• Determine the minimum and maximum, then find the median (Q2), the median of the first haf of the data,(Q1) then the median of the second half of the data(Q3)

Multiple Choice

What is the 5 number summary of this data set?

• 0, 2, 4, 5.5, 14

• 0, 1, 2.5, 9, 14

• 0, 1, 3.5, 6, 14

• 0, 1, 3, 6, 10

Mean

• The mean is also know as the average.

• To find the mean add together all of the data points and divide the sum by how many data points there are in the set.

•  $\frac{1+2+3+4+5}{5}=\frac{15}{5}=3,\ 3\ is\ the\ mean\ or\ average$  

Multiple Choice

What is the mean of this data set?

• 4.25

• 3.5

• 5

• 85

Range

• Range is found by subtracting the minimum from the maximum.

• So in a data set that looks like this: {23, 34, 45, 56, 78} you would subtract 23 from 78 (78 - 23)

• The range of the data is 55 (78-23 = 55)

Interquartile Range (IQR)

• Interquartile range (IQR) is the middle 50% of a data set.

• IQR is found by subtracting Q1 from Q3

• Q1 and Q3 of this set are in red (the median is in blue) {3,4,5,6,7,8,9}

• The IQR is found by 8-4

• The interquartile range is 4

Outliers

• An outlier is a data point that is WAY outside of most of the data.

• To find any technical outliers we need to first find the upper and lower fence.

• Lower fence is $Q_1-1.5\left(IQR\right)$  

• Upper Fence is  $Q_{3\ }+\ 1.5\left(IQR\right)$  

• Any data outside of the fences is an outlier. 

Multiple Choice

Are there any outliers this data set?

• Yes

• No

Open Ended

What are the lower and upper fences?

Open Ended

What was the outlier?

Multiple Choice

Which choice is the best description of the distribution data set?

• Uniform

• Symmetrical

• Skewed left

• Skewed right

Multiple Choice

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

• mean

• mode

• median

MAD Mean Absolute Deviation

• MAD shows the average difference the data set has from the mean or how far on average the data deviates from the mean.

• To find it, first find the mean.

• Then, find the absolute distance each data point has from the mean.

• Finally, find the mean of those absolute values.

MAD Mean Absolute Deviation

• {3,4,5,6,7,8,9}

• To find it, first find the mean.  $\frac{3+4+5+6+7+8+9}{6}=\frac{42}{7}=6$  

• Then, find the absolute distance each data point has from the mean. The absolute value of (3-6, 4-6, 5-6, 6-6, 7-6, 8-6, 9-6) or (3, 2, 1, 0, 2, 3)

• Finally, find the mean of those absolute values. $\frac{3+2+1+0+1+2+3}{6}=\frac{12}{2}=2$  

Make Sure:

• You've completed all of you quizzes! (Especially 6.11!)

• To do the test once now! (Part 1 and Part 2)

Questions?