Stats 3.1 central tendency
The population arithmetic mean
(U) pronounced mew, is computed using all the individuals in a population. The population mean is a parameter
Mean and median are close together in a bell shaped distribution and we would use the mean as the measure of central tendency.
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Note that N reps the size of the population and n reps the size of the sample.
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The mode can be computed for either quantitate or qualitative data (mean and median are only relative to quantitative data).
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A measure of central tendency numerically describes the average or typical data value.
3 most widely used measures of central tendency are the mean, median and mode.
Because extreme values don't affect the value of the median, we say it's resistant.
A numerical summary of data is said to be resistant if extreme values (very large or small) relative to the data don't affect its value substantially.
The mode is the most frequent observation of the variable that occurs in the data set.
A set of data can have no mode, one mode, or more than one mode.
Mean interpretation
Center of gravity. It's used when data are quantitative and the frequency distribution is roughly symmetric
Median interpretation
Divides the bottom 50% of the data from the top 50%. Used when the data are quantitative and the frequency distribution is skewed left or right.
If the number of observations is 10, arrange data in ascending order and use the following formula
Find the 2 middle numbers: n/2 = 10/2 = 5 n/2 + 1 = 10/2 + 1 = 6 So the 5th number is 77 and the 6th number is 82, find the meaning and you'll get the median M=77+82/2 = 79.5
Greek letters represent parameters and roman letters represent statistics.
Formula: if x1, x2...., xn are the N observations of a variable from a population, then the population mean, u, is u = x1 + x2 + ...... + xn/N = summation of xi/N
If the number of observations is even, then the median is the mean of the two middle observations in the data set.
That is the median is the mean of the two observations that lie in the n/2 position and the n/2+1 position.
The word Average often represents the arithmetic mean.
The arithmetic mean of a variable is computed adding all the values of the variable in the data set and dividing by the total number of observations.
Page 129 for calculator instructions on how to compute mean and median
The mean is sensitive to extreme values, the median is not.
If the number of observations is odd, then the median is the data value exactly in the middle of the data set.
The median lies in the (n+1)/2 position
The sample arithmetic mean, x (with bar over it and pronounced x-bar) is computed using sample data.
The sample mean is a statistic.
If there are 9 observations, to find the number in the middle use the following formula
n+1/2 = 9+1/2 = 5th data value, list the data in ascending order and choose the 5th value.
If x1, x2, ....., xn are n observations of a variable from a sample, then the sample mean, x, is
x = x1+x2+ .... + xn/n = summation of xi/n
Bimodal: if a data set has two pairs of values (77, 77, 88, 88) it is bimodal.
If a data set has three or more data values that occur with the highest frequency, the data set is multimodal. The mode is not reported for multimodal data because it's not representative of a typical value.
When data are skewed (left or right) there are extreme values in the tail which tend to pull the mean in the direction of the tail.
In skewed right distributions there are large observations in the right tail. These observations increase the value of the mean and it is a larger # than the median. If the data is skewed left the mean is smaller than the median. Symmetric distributions: mean and median are roughly equal.
The median
The median of a variable is the value that lies in the middle of the data when arranged in ascending order. M is used to represent the median.
