AP Statistics Curriculum 2007 EDA Center
Contents
General Advance-Placement (AP) Statistics Curriculum - Central Tendency
Measurements of Central Tendency
There are three main features of all populations (or data samples) that are always critical in understanding and interpreting their distributions. These characteristics are Center, Spread and Shape. The main measures of centrality are mean, median and mode.
Suppose we are interested in the long-jump performance of some students. We can carry an experiment by randomly selecting eight male statistics students and ask them to perform the standing long jump. In reality every student participated, but for the ease of calculations below we will focus on these eight students. The long jumps were as follows:
| 74 | 78 | 106 | 80 | 68 | 64 | 60 | 76 |
Mean
The sample-mean is the arithmetic average of a finite sample of numbers. For instance, the mean of the sample \(x_1, x_2, x_3, \cdots, x_{n-1}, x_n\) (short hand notation\[\{x_{i}\}_{i=1}^n\]), is given by:
In the long-jump example, the sample-mean is calculated as follows:
Median
The sample-median can be thought of as the point that divides a distribution in half (50/50). The following steps are used to find the sample-median:
- Arrange the data in ascending order
- If the sample size is odd, the median is the middle value of the ordered collection
- If the sample size is even, the median is the average of the middle two values in the ordered collection.
For the long-jump data above we have:
- Ordered data:
| 60 | 64 | 68 | 74 | 76 | 78 | 80 | 106 |
- \(Median = {74+76 \over 2} = 75\).
Mode(s)
The modes represent the most frequently occurring values (The numbers that appear the most). The term mode is applied both to probability distributions and to collections of experimental data.
For instance, for the Hot dogs data file, there appears to be three modes for the calorie variable. This is presented by the histogram of the Calorie content of all hotdogs, shown in the image below. Note the clear separation of the calories into three distinct sub-populations - the highest points in these three sub-populations are the three modes for these data.
Resistance
A statistic is said to be resistant if the value of the statistic is relatively unchanged by changes in a small portion of the data. Referencing the formulas for the median, mean and mode which statistic seems to be more resistant?
If you remove the student with the long jump distance of 106 and recalculate the median and mean, which one is altered less (therefore is more resistant)? Notice that the mean is very sensitive to outliers and atypical observations, and hence less resistant than the median.
The following two sample measures of population centrality estimate resemble the calculations of the mean, however they are much more resistant to change in the presence of outliers.
K-times trimmed mean
\(\bar{y}_{t,k}={1\over n-2k}\sum_{i=k+1}^{n-k}{y_{(i)}}\), where \(k\geq 0\) is the trim-factor (large k, yield less variant estimates of center), and \(y_{(i)}\) are the order statistics (small to large). That is, we remove the smallest and the largest k observations from the sample, before we compute the arithmetic average.
Winsorized k-times mean
The Winsorized k-times mean is defined similarly by \(\bar{y}_{w,k}={1\over n}( k\times y_{(k)}+\sum_{i=k+1}^{n-k}{y_{(i)}}+k\times y_{(n-k+1)})\), where \(k\geq 0\) is the trim-factor and \(y_{(i)}\) are the order statistics (small to large). In this case, before we compute the arithmetic average, we replace the k smallest and the k largest observations with the kth and (n-k)th largest observations, respectively.
Other Centrality Measures
The arithmetic mean answers the question, if all observations were equal, what would that value (center) have to be in order to achieve the same total? \[n\times \bar{x}=\sum_{i=1}^n{x_i}\]
In some situations, there is a need to think of the average in different terms, not in terms of arithmetic average.
Harmonic Mean
If we study speeds (velocities) the arithmetic mean is inappropriate. However the harmonic mean (computed differently) gives the most intuitive answer to what the "middle" is for a process. The harmonic mean answers the question if all the observations were equal, what would that value have to be in order to achieve the same sample sum of reciprocals?
- Harmonic mean\[\hat{\hat{x}}= \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3} + \ldots + \frac{1}{x_n}}\]
Geometric Mean
In contrast, the geometric mean answers the question if all the observations were equal, what would that value have to be in order to achieve the same sample product?
- Geometric mean\[\tilde{x}^n={\prod_{i=1}^n x_i}\]
- Alternatively\[\tilde{x}= \exp \left( \frac{1}{n} \sum_{i=1}^n\log(x_i) \right)\]
Medimean
We have already seen the standard definitions for mean and median. There is an alternative ways to define these measures of centrality using optimization theory.
- Median = argxmin(∑s∈S│s-x│1)
- Average = argxmin(∑s∈S│s-x│2)
- where \( \operator{\argmin}\) provides the value x which minimizes the distance function in the operator.
When we are looking for a median/average/whatever, we want a point that is fairly close to all the members in the sequence. In other words, a point for which the sum of distances from each member is the lowest. This holds true for both the median and the average. The critical difference between them is how we define distance.
When we take the average, we actually care about the Euclidean distance squared. (Even though we may not know about it when we calculate the average in the normal, simple way.) When we take the median, we care about the Euclidean distance itself, not squared.
What does it mean, in the case of average, that we care about the distance squared? It means that big distances have a lot more effect on the final number than small distances. Which in turn means that far-away members, like our two bosses, have a big influence on the average. So the average goes a considerable amount in their direction.
In the median though, we care about the distance itself, not squared. So far-away members cease to have the big influence they had on the average. As a result we get a number that is closer to the common employee.
Here are the definitions repeated:
Median(S) = argxmin(∑s∈S│s-x│1)
Average(S) = argxmin(∑s∈S│s-x│2)
You can see that they are the same, except for that exponent in the middle. For the median the exponent is 1, and for the average it is 2. (This is the whole business of distance^1 vs. distance^2 we talked about.) So average and median are actually the same animal, just with a different parameter!
1. The medimean
This leads us into the first “invention”: The medimean. The word medimean is a fusion of median and mean, (which is another word for average). This is the definition of the medimean:
Medimean(S, i) = argxmin(∑s∈S│s-x│i)
The medimean takes a parameter i which we can call “the distance exponent”. When given a distance exponent of 1, the medimean produces the median†. When given a distance exponent of 2, the medimean produces the average.
So the medimean is a continuous and natural transition between the median and the average for a given sequence of members:
What can the medimean be useful for? It may be useful when you want some sort of compromise between the average and the median. For example, if you are interested in something like a median, but which moves just a little bit in the direction of far-away elements, if there are any. In that case you may want to use a 1.1-medimean for your analysis.
Problems
References
- SOCR Home page: http://www.socr.ucla.edu
Translate this page:
