Descriptive Statistics, Correlation, and Regression: Introduction to Biostatistics, Lecture notes of Biostatistics

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Descriptive statistics; Correlation and regression

Patrick Breheny

September 16

Descriptive statistics CorrelationRegression Numerical summariesPercentiles

Tables and figures

Human beings are not good at sifting through large streams of data; we understand data much better when it is summarized for us We often display summary statistics in one of two ways: tables and figures Tables of summary statistics are very common (we have already seen several in this course) – nearly all published studies in medicine and public health contain a table of basic summary statistics describing their sample However, figures are usually better than tables in terms of distilling clear trends from large amounts of information

Descriptive statistics CorrelationRegression Numerical summariesPercentiles

Categorical data

Summarizing categorical data is pretty straightforward – you just count how many times each category occurs Instead of counts, we are often interested in percents A percent is a special type of rate, a rate per hundred Counts (also called frequencies), percents, and rates are the three basic summary statistics for categorical data, and are often displayed in tables or bar charts, as we saw in lab

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Continuous data

For continuous data, instead of a finite number of categories, observations can take on a potentially infinite number of values Summarizing continuous data is therefore much less straightforward To introduce concepts for describing and summarizing continuous data, we will look at data on infant mortality rates for 111 nations on three continents: Africa, Asia, and Europe

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Histogram of European infant mortality rates

Deaths per 1,000 births

Count

0 2 4

6

8 10

0 50 100 150 200

Africa

0 2

4 6

8 10

Asia

0 5 10 15 20 25

Europe

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Summarizing continuous data

As we can see, continuous data comes in a variety of shapes Nothing can replace seeing the picture, but if we had to summarize our data using just one or two numbers, how should we go about doing it? The aspect of the histogram we are usually most interested in is, “Where is its center?” This is typically represented by the average

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Spread

The second most important bit of information from the histogram to summarize is, “How spread out are the observations around the center”? This is most typically represented by the standard deviation To understand how standard deviation works, let’s return to our small example with the numbers { 4 , 5 , 1 , 9 } Each of these numbers deviates from the mean by some amount:

4 − 4 .75 = − 0. 75 5 − 4 .75 = 0. 25 1 − 4 .75 = − 3. 75 9 − 4 .75 = 4. 25

How should we measure the overall size of these deviations?

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Root-mean-square

Taking their mean isn’t going to tell us anything (why not?) We could take the average of their absolute values:

|− 0. 75 | + | 0. 25 | + |− 3. 75 | + | 4. 25 | 4

But it turns out that for a variety of reasons, the root-mean-square works better as a measure of overall size: √ (− 0 .75)^2 + (0.25)^2 + (− 3 .75)^2 + (4.25)^2 4

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Meaning of the standard deviation

The standard deviation (SD) describes how far away numbers in a list are from their average The SD is often used as a “plus or minus” number, as in “adult women tend to be about 5’4, plus or minus 3 inches” Most numbers (roughly 68%) will be within 1 SD away from the average Very few entries (roughly 5%) will be more than 2 SD away from the average This rule of thumb works very well for a wide variety of data; we’ll discuss where these numbers come from in a few weeks

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Standard deviation and the histogram

Background areas within 1 SD of the mean are shaded:

Deaths per 1,000 births

Count

0 2 4 6 8 10 50 100 150 200

Africa

0

2

4

6

0 50 100 150

Asia

0

5

10

15

10 20 30 40

Europe

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Summaries can be misleading!

All of the following have the same mean and standard deviation:

−4 −2 0 2 4

Frequency

−4 −2 0 2 4

−4 −2 0 2 4

Frequency

−4 −2 0 2 4

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Percentiles

The average and standard deviation are not the only ways to summarize continuous data Another type of summary is the percentile A number is the 25th percentile of a list of numbers if it is bigger than 25% of the numbers in the list The 50th percentile is given a special name: the median The median, like the mean, can be used to answer the question, “Where is the center of the histogram?”

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Skew

Note that the histogram for Europe is not symmetric: the tail of the distribution extends further to the right than it does to the left Such distributions are called skewed The distribution of infant mortality rates in Europe is said to be right skewed or skewed to the right For asymmetric/skewed data, the mean and the median will be different

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Hypothetical example

Azerbaijan had the highest infant mortality rate in Europe at 37 What if, instead of 37, it was 200? Mean Median Real 14.1 11 Hypothetical 19.2 11 The mean is now higher than 72% of the countries Note that the average is sensitive to extreme values, while the median is not; statisticians say that the median is robust to the presence of outlying observations