Measures of spread include:

the Inter-quartile Range (IQR)

the Standard Deviation

1. Find the mean.

2. Find the deviation for each value in the data set.

3. Square each deviation.

4. Add all the squared deviations together.

5. Divide the sum of the squared deviations by N-1, where N is the total number of values in the data set.

**This is the Variance**

6. Take the square root of the variance.

**This is the Standard Deviation**

**Displaying and Summarizing Quantitative Data**

Displaying Quantitative Data

Histograms

Stem and Leaf Displays

Dot Plots

Describing Quantitative Data

The Center of Attention

The center is the place in the distribution of a variable that you'd point to if you wanted to attempt the impossible by summarizing the entire distribution with a single number.

Measures of center include:

the Mean

the Median

**Putting It All Together**

When describing the distribution of a quantitative data set, remember to report:

I. Shape

-modes or uniform

-skewed or symmetric

-outliers and gaps

II. Center

-the mean is usually used as the measure of center for histograms that are symmetric and have no outliers

-the median is usually used as the measure of center for histograms that are skewed or have an outlier

III. The Spread

-The IQR = Q3 - Q1 ; usually used with the median

-The standard deviation is the square root of the variance; usually used with the mean

A histogram uses adjacent bars to show the distribution of a quantitative variable. Each bar represents the frequency (or relative frequency) of values falling within each bin.

Histograms

A stem and leaf display show quantitative data values in a way that sketches the distribution of the data.

Stem and Leaf Displays

A dot plot graphs a dot for each case against a single axis.

Dot Plots

WAIT!!! Think About It

Remember to check for the

Quantitative Data Condition: The data are values of a quantitative variable whose units are known.

What this means:

We can not use a histogram, stem and leaf display, or a dot plot to display CATEGORICAL DATA.

When describing the shape of the distribution, look for

single or multiple modes

symmetry or skewness

outliers and gaps

The Shape of the Distribution

Look for Humps

The mode is a HUMP or local HIGH point in the shape of the distribution.

The mode can be

unimodal- one mode

bimodal- two modes

multimodal- more than two modes

There can also be no mode.

Look for the HUMPS

A histogram that does not have a mode and the bars appear to all be approximately the same height is called uniform.

NO Humps?

Does the Distribution have Symmetry?

Can you draw a vertical line down the middle of the histogram, fold it in half, and have an almost perfect match?

YES?

Then the distribution is SYMMETRIC.

NO?

The thinner ends of a distribution are called TAILS.

If one tail stretches out farther than the other, the histogram is said to be SKEWED to the longer side of the tail.

Do You Have a Tail???

When describing the DISTRIBUTION of quantitative data you should always tell about these three things:

1. The Shape

2. The Center

3. The Spread

The Outsiders

Do any unusual features stick out in the histogram?

Are there any gaps in the data?

Are there any outliers?

Outliers are the stragglers that stand off away from the body of the distribution.

The Mean Mean

The mean is used by summing (adding) all the data values together and dividing by the count.

The formula to the right, pronounced x bar, is the statistical formula for mean.

The Median in the Middle

The median is the middle value, with half of the data above and half the data below it.

In order to find the median of a batch of n numbers,

1. Put the n numbers in numerical order from least to greatest.

2. If n is odd, the median is the middle value.

3. If n is even, there are two middle values. So in this case, we take the average of the two middle values.

All Spread Out

The Inter-Quartile Range

The Quartiles: Q1, Q2, Q3, Q4

The lower quartile (Q1) is the value with one quarter of the data below it.

The upper quartile (Q3) has three quarters of the data below it.

The median (Q2) and the quartiles divide the data into four parts with equal numbers of data values.

The Inter Quartile Range tells us how much data falls in the middle of the half of the data

IQR = Q3-Q1

How to find the Variance and the Standard Deviation

Standard Deviation

The standard deviation, denoted s, is the the square root of the Variance.

The variation is the sum of squared deviations from the mean, divided by the count minus 1.

The standard deviation is usually reported with the mean

Variance Formula

Formula for Standard Deviation

The Mean

-the mean is usually reported with the standard deviation

-the mean is the balancing point in the histogram

The Median

-the median is usually reported with the IQR

-the median and IQR are resistant to outliers and skewness