Chapter 3: Numerical Summaries for Quantitative Data

Mode

The most frequently occurring data value in a set

Is the Mode a good measure of center? Why?

No, because it can fall anywhere in the distribution

What are the two most common ways to measure the center of a distribution?

The Median and Mean

Median

The midpoint of a distribution - the number such that about half the observations are smaller and about half are larger

How to find the Median of a distirbution?

Arrange the data values from smallest to largest and find the midpoint

Mean

The average of all the individual data values

The notation x̄ refers to the ____ of a ____.
The notation μ refers to the ____ of a ____.

Mean; sample
mean; population

Resistant

A statistic that is not affected much by outliers

The Mean is ____ to extreme values such as outliers.

Not resistant

The Median is _____ to extreme values such as outliers.

Resistant

When measuring the center, when do you use Median and when do you use Mean?

If the quantitative data is roughly symmetric and has no outliers, use the Mean. It the data is strongly skewed or has outliers, use the Median.

True/False: The Median of a quantitative data set is one of the individual data values

False

What are the three most common ways to measure the variability of a distribution of quantitative data?

The range, standard deviation, and interquartile range

Define Range

The distance between the minimum and maximum value

Range = maximum - minimum

The range of a data set is a ____ _____.

single number

Is the Range a resistant measure of variability?

No

What does Standard Deviation measure?

Measures the typical distance of the values in a distribution from the Mean

What are 5 steps to find the Sample Standard Deviation?

1. Find the Mean of the distribution

2. Calculate the deviation of each value from the mean: deviation = value-mean

3. Square each deviation

4. Add all the squared deviations and divide by n-1

   1. Take the square root

___ refers to the Standard Deviation for a sample.
___ refers to the Standard Deviation for a population.

S_x ; σ

The value obtained before taking the square root of a standard deviation is call the _____.

Variance (S_x²)

What does it mean when S_x= 0?

There is no variability

Is S_x resistant?

No

S_x is only when the ____ is the chosen measure of center.

Mean

How to find Quartiles?

Arrange the data values from left to right, smallest to largest and find the Median, then separate into four groups with roughly the same number of values

First Quartile (Q₁)

The Median of the data values that are to the left

Third Quartile (Q₃)

The Median of the data values that are to the right

Interquartile Range (IQR)

The distance between the first and third Quartiles of a distribution

Are the quartiles and Interquartile Range resistant?

Yes

The ____ and ____ are usually better choices than the ____ and ____ for describing a skewed distribution or one with outliers.

median; IQR
mean; standard deviation

Use the ____ and ____ for roughly symmetric distributions that don’t have outliers.

mean; standard deviation

What is often used as a ruler for identifying outliers?

Interquartile range (IQR)

How to identify outliers with The 1.5 x IQR Rule?

A data point is an outlier if it falls more than 1.5 x IQR above the third quartile or below the first quartile.

low outliers < Q₁ - 1.5 x IQR high outliers > Q₃ + 1.5 x IQR

What are three reasons as to why it’s important to identify outliers?

1. They might be inaccurate data values

2. They can indicate a remarkable occurrence

3. They can heavily influence the values of some summary statistics

Five-Number Summary

Consists of the minimum, the first quartile, the median, the third quartile, and the maximum

Boxplot

A visual representation of the five-number summary

What is another name for a Boxplot?

A Box-and-Whisker Plot

What are 8 steps to making a Boxplot?

1. Find the five-number summary for the distribution

2. Identify outliers using the 1.5 x IQR rule

3. Draw & label the axes

4. Scale the axes

5. Draw a box

6. Mark the median with a vertical line segment that’s the same height as the box

7. Mark any outliers with a special symbol

8. Draw whiskers that extend from the ends of the box to the smallest and largest data values that are NOT outliers

Percentile

The percentage of values in a distribution that are less than the individual’s data value

Percentile’s are used if a quantitative data set contains _______.

A large number of values

Which state is at the 78th percentile of a distirbution?

The one with (0.78)(50)=39 data values less than its own percentage.

Standardized Score (Z-Score)

The score for an individual in a distribution that tells us how many standard deviations away from the mean the value falls, and in which direction

Correlation “r”

A measure of the strength and direction of a linear relationship between two quantitative variables

What is the Correlation “r” sometimes referred to as?

The correlation coefficient

The extreme values, ____ and ____ occur only in the case of a perfect linear relationship, when the points lie exactly along a straight line.

r = -1; r = 1

If a linear relationship is weak, the Correlation “r” will be ______.

Close to 0

T/F: A Correlation “r” close to 1 or -1 implies that an association is linear.

False

Correlation “r” alone doesn’t provide information about ____.

Form (linear or non-linear)

T/F: Correlation “r” does not imply causation.

True

To calculate the Correlation “r”, both variables much be ______.

Quantitative

T/F: Correlation “r” makes no distinction between explanatory and responsive variables.

True

Does the Correlation “r” change when we change the variables’ units? Why?

No. The correlation has no units of measurement because it’s found using the z-score

The Correlation “r” is a resistant/not resistant measure of strength?

Not resistant