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Displaying and Summarizing Quantitative Data

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by

T Long

on 14 January 2014

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Transcript of Displaying and Summarizing Quantitative Data

The spread is a numerical summary of how tightly the values are clustered around the center.

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
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