Chapter 2

Describing distributions with numbers

Chapter Outline

1. Measuring center: the mean 2. Measuring center: the median 3. Comparing the mean and the median 4. Measuring spread: the quartiles 5. The five-number summary and boxplots 6. Measuring spread: the standard deviation 7. Choosing measures of center and spread

Measuring center: the mean

Notation: It is simply the ordinary arithmetic

average. Suppose that we have n observations

(data size, number of individuals). Observations are denoted as x1, x2, x3,

…xn.

x

Measuring center: the mean

How to get ?

Example 2.1 (P.33)

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Measuring center: the median

Notation: M Median M is the midpointmidpoint of a

distribution half the observations are smaller than M and the other half are larger than M.

Measuring center: the median

How to find M?

– 1. Sort all observations in increasing order Sort all observations in increasing order (This step is important!!!)(This step is important!!!)

– 2. If n is odd, observation is M. if n is even, average of two center values is M.

Note that is the location of the median in the ordered list, not the median value.

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Measuring center: the median

Examples

Case 1. 11, 21, 13, 24, 15, 26, 17 Case 2. 11, 21, 13, 24, 15, 26

Example 2.2, 2.3 (P.35)

Mean vs. Median

Median is more resistant than the mean. The mean and median of a symmetric

distribution are close together. If the distribution is exactly symmetric, the mean and median are exactly the same. In a skewed distribution, the mean is farther out in the long tail than is the median.

Example1, 2, 3, 4, 5, 6, 10000

Inference :

Strongly skewed distributions are reported with median than the mean.

Measuring Spread: The Quartiles

The quartiles mark out the middle half of the distribution.

Calculating the Quartiles :

– Step1. Arrange the observations in increasing order and

locate the median M in the ordered list of observations.

– Step2. The first quartile Q1 is the median of the

observations whose position in the ordered list is to the left of the location of the overall median.

– Step3. The third quartile Q3 is the median of the

observations whose position in the ordered list is to the right of the location of the overall median.

Measuring spread: the quartiles

Example 2.4 (P. 37) Example 2.5 (P. 38)

Note:

(1) It is important to sort data first before

we try to find quartiles!

(2) Quartiles are resistant.

The five-number summary and boxplots

The five-number summary:

Minimum, Q1, M, Q3, Maximum.

Boxplot is a graph of five number

summary.

Boxplots are most useful for side-by-side comparison of several distributions.

Boxplot

1. A boxplot is a graph of the five-number summary

2. A central box spans the quartiles 3. A line in the box marks the median 4. Lines extended from the box out to

the minimum and maximum 5. Range = maximum - minimum

The five-number summary and boxplot

Figure 2.2(P.39): side-by-side boxplots comparing the distributions of earning for two levels of education.

The five-number summary and boxplots

Inference :

Boxplot also gives an indication of the symmetry or skewness of a distribution.

-- In a symmetric distribution Q1 and Q3 are equally distant from the median, but in case of right skewed one the third quartile would be further above the median than the first quartile bellow it.

Measuring spread: the standard deviation

It says how far the observations are from their mean. The variance s2 of a set of observations is an average of the

squares of the deviations of the observations from their mean. Notation: s2 for variance and s for standard deviation

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Why (n-1) ?

As the sum of the deviations always equals 0, so the knowledge of (n-1) of

them determines the last one. --- Only (n-1) of the squared deviations are

variable but not the last one, so we average by dividing the total by (n-1).

The number (n-1) is called the degrees of freedom of the variance or standard deviation

)( xxi

Measuring spread: the standard deviation

To find the variance and the standard deviation– 1. Find the mean of the data set – 2. Subtract the mean from each number (we call

that deviation)– 3. Square each result– 4. Sum all the square– 5. Divide the sum of square by n-1, where n is the

number of all observations. Now you get variance– 6. Standard deviation is just the positive square

root of the variance.

Measuring spread: the standard deviation

Example 2.6 (P.42)

Properties of s2 and s

s measures spread about the mean and should be used only when the mean is chosen as the measure of center.

s 0 and s=0 only when each of the observation values does not differ from each other.

S is not resistant.

Choosing measures of center and spread

With a skewed distribution or with a distribution with extreme outliers, five-number summary is better.

With a symmetric distribution (without outliers), mean and standard deviation are better.