Measures of Center
CO-4: Distinguish among different measurement scales, choose the appropriate descriptive and inferential statistical methods based on these distinctions, and interpret the results.
LO 4.4: Using appropriate graphical displays and/or numerical measures, describe the distribution of a quantitative variable in context: a) describe the overall pattern, b) describe striking deviations from the pattern
LO 4.7: Define and describe the features of the distribution of one quantitative variable (shape, center, spread, outliers).
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Intuitively speaking, a numerical measure of center describes a “typical value” of the distribution.
The two main numerical measures for the center of a distribution are the mean and the median.
In this unit on Exploratory Data Analysis, we will be calculating these results based upon a sample and so we will often emphasize that the values calculated are the sample mean and sample median.
Each one of these measures is based on a completely different idea of describing the center of a distribution.
We will first present each one of the measures, and then compare their properties.
LO 4.8: Define and calculate the sample mean of a quantitative variable.
The mean is the average of a set of observations (i.e., the sum of the observations divided by the number of observations).
The mean is the average of a set of observations
- The sum of the observations divided by the number of observations).
- If the n observations are written as
- their mean can be written mathematically as:their mean is:
We read the symbol as “x-bar.” The bar notation is commonly used to represent the sample mean, i.e. the mean of the sample.
Using any appropriate letter to represent the variable (x, y, etc.), we can indicate the sample mean of this variable by adding a bar over the variable notation.
EXAMPLE: Best Actress Oscar Winners
We will continue with the Best Actress Oscar winners example (Link to the Best Actress Oscar Winners data).
34 34 26 37 42 41 35 31 41 33 30 74 33 49 38 61 21 41 26 80 43 29 33 35 45 49 39 34 26 25 35 33
The mean age of the 32 actresses is:
We add all of the ages to get 1233 and divide by the number of ages which was 32 to get 38.5.
We denote this result as x-bar and called the sample mean.
Note that the sample mean gives a measure of center which is higher than our approximation of the center from looking at the histogram (which was 35). The reason for this will be clear soon.
EXAMPLE: World Cup Soccer
Often we have large sets of data and use a frequency table to display the data more efficiently.
Data were collected from the last three World Cup soccer tournaments. A total of 192 games were played. The table below lists the number of goals scored per game (not including any goals scored in shootouts).
|Total # Goals/Game
To find the mean number of goals scored per game, we would need to find the sum of all 192 numbers, and then divide that sum by 192.
Rather than add 192 numbers, we use the fact that the same numbers appear many times. For example, the number 0 appears 17 times, the number 1 appears 45 times, the number 2 appears 51 times, etc.
If we add up 17 zeros, we get 0. If we add up 45 ones, we get 45. If we add up 51 twos, we get 102. Repeated addition is multiplication.
Thus, the sum of the 192 numbers
= 0(17) + 1(45) + 2(51) + 3(37) + 4(25) + 5(11) + 6(3) + 7(2) + 8(1) = 453.
The sample mean is then 453 / 192 = 2.359.
Note that, in this example, the values of 1, 2, and 3 are the most common and our average falls in this range representing the bulk of the data.
LO 4.9: Define and calculate the sample median of a quantitative variable.
The median M is the midpoint of the distribution. It is the number such that half of the observations fall above, and half fall below.
To find the median:
- Order the data from smallest to largest.
- Consider whether n, the number of observations, is even or odd.
- If n is odd, the median M is the center observation in the ordered list. This observation is the one “sitting” in the (n + 1) / 2 spot in the ordered list.
- If n is even, the median M is the mean of the two center observations in the ordered list. These two observations are the ones “sitting” in the (n / 2) and (n / 2) + 1 spots in the ordered list.
EXAMPLE: Median (1)
For a simple visualization of the location of the median, consider the following two simple cases of n = 7 and n = 8 ordered observations, with each observation represented by a solid circle:
- In the images above, the dots are equally spaced, this need not indicate the data values are actually equally spaced as we are only interested in listing them in order.
- In fact, in the above pictures, two subsequent dots could have exactly the same value.
- It is clear that the value of the median will be in the same position regardless of the distance between data values.
EXAMPLE: Median (2)
To find the median age of the Best Actress Oscar winners, we first need to order the data.
It would be useful, then, to use the stemplot, a diagram in which the data are already ordered.
- Here n = 32 (an even number), so the median M, will be the mean of the two center observations
- These are located at the (n / 2) = 32 / 2 = 16th and (n / 2) + 1 = (32 / 2) + 1 = 17th
Counting from the top, we find that:
- the 16th ranked observation is 35
- the 17th ranked observation also happens to be 35
Therefore, the median M = (35 + 35) / 2 = 35
Comparing the Mean and the Median
LO 4.10: Choose the appropriate measures for a quantitative variable based upon the shape of the distribution.
As we have seen, the mean and the median, the most common measures of center, each describe the center of a distribution of values in a different way.
- The mean describes the center as an average value, in which the actual valuesof the data points play an important role.
- The median, on the other hand, locates the middle value as the center, and the orderof the data is the key.
To get a deeper understanding of the differences between these two measures of center, consider the following example. Here are two datasets:
|Data set A → 64 65 66 68 70 71 73
|Data set B → 64 65 66 68 70 71 730
For dataset A, the mean is 68.1, and the median is 68.
Looking at dataset B, notice that all of the observations except the last one are close together. The observation 730 is very large, and is certainly an outlier.
In this case, the median is still 68, but the mean will be influenced by the high outlier, and shifted up to 162.
The message that we should take from this example is:
The mean is very sensitive to outliers (because it factors in their magnitude), while the median is resistant (or robust) to outliers.
- For symmetric distributions with no outliers: the mean is approximately equal to the median.
- For skewed right distributions and/or datasets with high outliers: the mean is greater than the median.
- For skewed left distributions and/or datasets with low outliers: the mean is less than the median.
Conclusions… When to use which measures?
- Use the sample mean as a measure of center for symmetric distributions with no outliers.
- Otherwise, the median will be a more appropriate measure of the center of our data.
- The two main numerical measures for the center of a distribution are the mean and the median. The mean is the average value, while the median is the middle value.
- The mean is very sensitive to outliers (as it factors in their magnitude), while the median is resistant to outliers.
- The mean is an appropriate measure of center for symmetric distributions with no outliers. In all other cases, the median is often a better measure of the center of the distribution.