STATISTICS - CLUTCH CH.2: MEASURES OF CENTER AND...

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STATISTICS - CLUTCH

CH.2: MEASURES OF CENTER AND SPREAD

MEAN & MEDIAN

Knowing the measures of center and spread let you compare one distribution to another

(1) MEAN = _______________ (2) MEDIAN = _______________

∑

EXAMPLE 1: Find the mean and median for the following data set:

12 18 32 15 18 20 24

EXAMPLE 2: Find the mean and median for the following data set:

12 18 32 15 18 20 24 100

MEAN VERSUS MEDIAN

Measures can be ___________, or resistant to outliers

The mean gets pulled toward _____________

The median resists outliers and maintains its central tendency

SKEWED TO THE __________ SYMMETRIC SKEWED TO THE __________

___ ___ ___ ___ ___ ___

EXAMPLE 3: What does the result of Example 2 tell you about the shape of the distribution? Which measure is more

appropriate to describe its center, the mean or the median?

n = sample

size

xi = each

observation

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PRACTICE 1: Determine the mean and median for the following data set:

81 8 4 2 3 1 2 9 9 2

PRACTICE 2: Determine the mean and median for the following data set:

7 56 46 32 69 34 53 76

PRACTICE 3: What can be said about the data from Practice 1?

PRACTICE 4: What can be said about the data from Practice 2?

PRACTICE 5: Which measure of central tendency is more appropriate for the data from Practice 1?

PRACTICE 6: Which measure of central tendency is more appropriate for the data from Practice 2?

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PROPERTIES OF THE SD

A higher standard deviation (______) tells you that the observations are further away from their corresponding mean

It follows that a standard deviation of zero means that all the observations are _________

If the mean is being used as the measure of the center of a distribution, the corresponding measure of spread is the

standard deviation or ____

The best situation to use the standard deviation is when the distribution is symmetric with no

DETERMINATION

The second version of the equation is easier for higher sample sizes

The variance is another measure not as commonly used, but can be found by ______

EXAMPLE 1: Determine the mean and standard deviation

of the following data set:

4 5 5 6 4 3 3 2

EXAMPLE 2: Determine the mean and standard deviation

of the following data set:

1 1 2 4 4 6 7 7

EXAMPLE 3: Which of the two previous examples (Example 1 or 2) are more appropriately described by the standard

deviation?

Step 1: Create a column with all the data

Step 2: Create another column with every xi-x

Step 3: Create a final column that squares each

value in the xi-x column

Step 4: Add up the values in the final column

Step 1: Create a column with all the data

Step 2: Create another column with every xi2

Step 3: Add up the values in the first column

Step 4: Add up the values in the second

column

n = sample

size

xi = each

observation √

∑ xi x

2

√

(∑ xi

2 (∑ xi)2

)

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PRACTICE 1: Determine the standard deviation for the following data set:

81 8 4 2 3 1 2 9 9 2 PRACTICE 2: Determine the standard deviation for the following data set:

7 56 46 32 69 34 53 76 PRACTICE 3: Which of the data sets (Practice 1 or Practice 2) is more spread out? PRACTICE 4: Which of the data sets (Practice 1 or Practice 2) is more appropriately described by the standard deviation?

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QUARTILES

Just as the name says, quartiles break data up into quarters or ______%, all based on the concept of medians

The median splits data up in half (______% above and below)

These quartiles then end up splitting up the data below and above the ______________ in half again

6

____ ____ ____

Reminder: the data needs to be placed in numerical order before determining any of this information

EXAMPLE 1: Determine the Q1 and Q3 for the following data set:

12 10 2 27 5 18 4

EXAMPLE 2: Determine the Q1 and Q3 for the following data set:

520 570 600 610 650 670 690 730

7 8 9 10 11

1 2 3 4 5

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INTERQUARTILE RANGE

A higher interquartile range (______) tells you that the observations are further away from their corresponding median

It follows that an IQR of zero means that all the observations are all __________

If the median is being used as the measure of the center of a distribution, the corresponding measure of spread is the IQR

The best situation to use the IQR is when the distribution is ___________________ or has extreme outliers

The IQR is determined by the following:

IQR = _____ – _____

FIVE-NUMBER SUMMARY

The five number summary is a simple way of getting a numerical picture of any distribution

(1) ______________ (2) Q1 (3) M (4) Q3 (5) ______________

EXAMPLE 1: Determine the five-number summary and interquartile range of the following data set:

15 80 24 56 67 34

Min: ____

Q1: ____

M: ____

Q3: ____

Max: ____

IQR: ____

DETECTING OUTLIERS

The most commonly used rule to detect and outlier is to use the _______

Lower outliers are any numbers lower than _________ below _____

Upper outliers are those higher than _________ above _____

EXAMPLE 2: Determine if there are any outliers from Example 1.

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BOXPLOTS

All you need is the five-number summary and the outliers of any data set in order to create its boxplot

The fences below Q1 and above Q3 only stop at the most extreme data point within the total _________ limit

____ ____ ____ ____ ____

_________ _________ _________

EXAMPLE 1: Construct the boxplot for the following data set:

30 32 52 61 34 102 0 42

Symmetry or _______________ can be determined by looking at distances from the minimum and Q1 to median and

comparing them to those from the maximum and Q3 to the median.

The further Q1 and the minimum are from the median, the data is likely skewed ___________

The further Q3 and the maximum are from the median, the data is likely skewed ___________

EXAMPLE 2: Looking at the boxplot from Example 1, what can be said about the distribution?

EXAMPLE 3: Looking at the boxplot below, what can be said about the distribution?

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PRACTICE 1: Determine the five number summary of the following data set and detect any outliers:

12 18 32 15 18 20 24 100

PRACTICE 2: Draw the boxplot for the data set in Practice 1. PRACTICE 3: From the boxplot in Practice 2, what can be said about the data set in Practice 1? PRACTICE 4: What can be said about the data set that generated the boxplot below?

PRACTICE 5: What can be said about the data set that generated the boxplot below?

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STATISTICS - CLUTCH CH.2: MEASURES OF CENTER AND...

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