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An Introduction to Statistics

Review of Statistics Lessons 3 and 4

Lesson 3: Measures of Central Tendancy

  1. Average is an ambiguous term referring often but not exclusively to the arithmetic mean.
  2. By average we usually mean some measure of central tendancy.
  3. Mode, median, and midrange are additional common averages.
  4. We find the arithmetic mean by summing all elements and dividing by the number of elements.
  5. Although x-bar is used for sample mean, µ (mu) is used for population mean.
  6. Sample size is the number of elements and is denoted by n.
  7. The population size is typically denoted by N.
  8. Mode is the data element which occurs most frequently.
  9. A uniform distribution can be said to have no mode.
  10. Distributions may also be bimodal or multimodal.
  11. The median is the middle element in an ordered data set.
  12. When there are an even number of elements, the median is the arithmetic mean of the middle two.
  13. The midrange is the arithmetic mean of the highest and lowest data elements.
  14. Do not confuse midrange, a measure of central tendancy, with range, a measure of dispersion.
  15. The mean is reliable (uses every data element) but can be distorted by outliers.
  16. While no average is the best, under certain circumstances one may be better than another.
  17. We typically report the mean to one more significant digit than the data.
  18. One should probably report the mean and standard deviation to the same precision.
  19. Another common rule in science is to use three significant digits (slide rule accuracy).

Lesson 4: Various Means

  1. The arithmetic mean is the sum of all elements divided by the number of elements.
  2. The geometric mean is used to find average rates of growth.
  3. The geometric mean is the nth root of the product of the data elements.
  4. nth roots can be found on your calculator using fractional exponents (½ would be square root).
  5. The harmonic mean is used to calculate average rates like speed.
  6. Harmonic mean is found by dividing n by the sum of reciprocals of the data elements.
  7. Reciprocal means "1 over the value".
  8. Speed is a scalar, whereas velocity is a vector (has both magnitude and direction).
  9. The quadratic mean is also known as Root Mean Square (RMS).
  10. It is used for AC voltage and is the square root of (the sum of the squares divided by n).
  11. The arithmetic mean of AC voltage is zero.
  12. The 10% trimmed mean is the arithmetic mean without the top 10% and bottom 10%.
  13. This avoids outlier distortion and corrects some skew.
  14. A distribution is skewed to the right if the mean is to the right of the median.
  15. Weighted means are most commonly encountered in GPA's where items have differing affects.
  16. Sometimes none of these means suffice and some combination is required.
  17. Be sure you can calculate means not only from a table of values, but also from a frequency table.