Five minutes late starting today. Chapter 3, some of Chapter 4, maybe. Chapter 3: Measures of Central Tendency
Various types of central tendency measures.
- Mode — The number which occurs most often in a set of data. A data set can have multiple modes. The mode is a poor method of finding the center of a data set, in many instances. If the data is grouped, the modal class is the class with the most members.
- Median — The number with as many data points larger than it as smaller than it. The middle number in a set of ranked data. The median isn’t always as well defined as we would like. If the data set contains an odd number of member, the middle member is the median. If the data set contains an even number of members, the median is some value between the two cetermost elements. By convention, the average of the two centermost elements is considered the median, but any number between those elements is a valid median. In grouped data, the median class is the class containing the median element, but often the exact median will be unknown; only the class it is located in will be certain. Linear interpolation is sometimes used to approximate an exact mode in grouped data.
- Mean — AKA the arithmetic mean, AKA the average. Divide sum of elements by number of elements. When discussing a population, the mean is a parameter of the population, and is represented by the greek letter mu. When discussing a sample, the mean is represented by x-bar, an x with a bar over the top. When dealing with grouped data, multiply the frequency of points in each class by the class mark, and average the ruslting data to get the mean. Mean is used a lot in statitics. Has good mathematical properties, is easy to calculate. Mean can be skewed by extreme data, however.
- Geometric Mean
- Harmonic Mean
- Percentiles and Quartiles
There are also various measures of variability
- Range and Interquartile Range — Range is equal to the maximum value minus the minimum value. Has some value for discussing variance but some problems. Range essentially only uses two values of data set. Also complicated to work with, mathematically. Empirical Rule states that the range is equal to six standard deviations. Normally the range is not calculated for grouped data, only for discrete data elements.
- Mean Absolute Deviation (MAD) — Mean Absolute Deviation is the average of the absolute values of the deviations around the mean for a set of numbers. Deviation from the mean is determined by subtracting the overall mean from each data element. Major problem with MAD is that it’s ugly; lots of work to generate, and a point of discontinuity in the derivatives of the function. Not used much.
- Variance (Standard Deviation is the square root of the variance) — Developed as a way of getting around the mathematical issues of the MAD. The Variance is the average of the squared deviations about the arithmetic mean for a set of numbers. The variance is the most widely used measure in statistics, because it is usable and mathematically convenient. The largest problem with it is that you end up wth squared units as the label of the answers. Often you want real units, not squared units. Standard deviation is the square root of the variance. Standard variation has handier units, but is ugly math again.
- Coefficient of Variation — Used primarily in finance, especially when finding rates of return.
Various measures of shape
- Skew — Skewness is when a distribution is asymmetrical or lacks symetry. Normal symmetry is a bell curve, aka normal distribution. An unbalanced graph will have a longer tail on one side of the mean. A curve with a long tail to the right side of the mean is positively skewed. A curve with a long tail to the left of the mean is negatively skewed.
- Kurtosis (peakedness) — See Pg. 78, generally how extreme the peak is.
Measures of Association
- Correlation Coefficient