The mean will likely be larger because the extreme values in the right tail tend to pull the mean in the direction of the tail.
*When data are either skewed left or skewed right, there are extreme values in the tail, which tend to pull the mean in the direction of the tail. If the distribution of the data is skewed right, there are large observations in the right tail. These observations tend to increase the value of the mean, while having little effect on the median.median.**
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This chapter concentrated on describing distributions numerically. Measures of central tendency are used to indicate the typical value in a distribution. Three measures of central tendency were discussed.
The mean measures the center of gravity of the distribution. The data must be quantitative to compute the mean.
The median separates the bottom 50% of the data from the top 50%. The data must be at least ordinal to compute the median.
The mode measures the most frequent observation. The data can be either quantitative or qualitative to compute the mode.
The median is resistant to extreme values, while the mean is not.
Measures of dispersion describe the spread in the data.
The range is the difference between the highest and lowest data values.
The standard deviation is based on the average squared deviation about the mean.
The variance is the square of the the standard deviation.
The range, standard deviation, and variance, are not resistant.
The mean and standard deviation are used in many types of statistical inference.
The mean, median, and mode can be approximated from grouped data.
The standard deviation can also be approximated from grouped data.
We can determine the relative position of an observation in a data set using z-scores and percentiles. A z-score denotes how many standard deviations an observation is from the mean.
Percentiles determine the percent of observations that lie above and below an observation.
The interquartile range is a resistant measure of dispersion. The upper and lower fences can be used to identify potential outliers. Any potential outlier must be investigated to determine whether it was the result of a data-entry error or some other error in the data-collection process, or is an unusual value in the data set.
The five-number summary provides an idea about the center and spread of a data set through the median and the interquartile range. The length of the tails in the distribution can be determined from the smallest and largest data values. The five-number summary is used to construct boxplots. Boxplots can be used to describe the shape of the distribution and to visualize outliers