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Univariate Equations

Module by: Adan Galvan. E-mail the author

Summary: Equations for univariate data.

Formulas for Univariate Data

Measures of Central Tendency

Mean

m=∑ X XN m X X N
(1)

Here mm is the population mean and NN is the number of scores. If the scores are from a sample, then the symbol MM refers to the mean and NN refers to the sample size. The formula for MM is the same as the formula for mm. The mean is a good measure of central tendency for roughly symmetric distributions but can be misleading in skewed distributions since it can be greatly influenced by extreme scores.

Geometric Mean

Geometric Mean=∏ X Xn Geometric Mean X X n
(2)

∏ X X X X means to take the product of all the values of XX. nn is the number of items being multiplied. The geometric mean can also be computed by:

  1. taking the logarithm of each number
  2. computing the arithmetic mean of the logarithms
  3. raising the base used to take the logarithms to the arithmetic mean.

Harmonic Mean

The harmonic mean is used to take the mean of sample sizes. If there are kk samples each of size nn, then the harmonic mean is defined as:

n h =k1 n 1 +1 n 2 +…+1 n k n h k 1 n 1 1 n 2 … 1 n k
(3)

Median

When there is an odd number of numbers, the median is simply the middle number. For example, the median of 2, 4, and 7 is 4. When there is an even number of numbers, the median is the mean of the two middle numbers. Thus, the median of the numbers 2, 4, 7, 12 is 4+72=5.5 4 7 2 5.5 .

Mode

The mode is the most frequently occurring score in a distribution and is used as a measure of central tendency.

Trimean

The trimean is computed by adding the 25th percentile plus twice the 50th percentile plus the 75th percentile and dividing by four.

Trimmed Mean

A trimmed mean is calculated by discarding a certain percentage of the lowest and the highest scores and then computing the mean of the remaining scores. For example, a mean trimmed 50% is computed by discarding the lower and higher 25% of the scores and taking the mean of the remaining scores. The median is the mean trimmed 100% and the arithmetic mean is the mean trimmed 0%.

Measures of Spread

Range

The range is the simplest measure of spread or dispersion: It is equal to the difference between the largest and the smallest values.

Semi-interquartile Range

The semi-interquartile range is a measure of spread or dispersion. It is computed as one half the difference between the 75th percentile [often called ( Q3Q3)] and the 25th percentile ( Q1Q1). The formula for semi-interquartile range is therefore:

Q1−Q22 Q1 Q2 2
(4)

Variance and Standard Deviation

The variance in a population is:

σ2=∑ X X−μN σ 2 X X μ N
(5)
where μμ is the mean and NN is the number of scores. When the variance is computed in a sample, the statistic
S2=∑ X X−MN S 2 X X M N
(6)
(where MM is the mean of the sample) can be used. S2 S 2 is a biased estimate of s2 s 2 , however. By far the most common formula for computing variance in a sample is:
s2=∑ X X−MN−1 s 2 X X M N 1
(7)
which gives an unbiased estimate of s2 s 2 . Since samples are usually used to estimate parameters, s2 s 2 is the most commonly used measure of variance.

The standard deviation is the square root of the variance. It is the most commonly used measure of spread.

Shape

Skew

Skew can be calculated as:

∑ X X−μ3N⁢σ3 X X μ 3 N σ 3
(8)

Kurtosis

Kurtosis is based on the size of a distribution's tails. Distributions with relatively large tails are called "leptokurtic"; those with small tails are called "platykurtic." A distribution with the same kurtosis as the normal distribution is called "mesokurtic." The following formula can be used to calculate kurtosis:

Kurtosis=∑ X X−μ4N⁢σ4−3 Kurtosis X X μ 4 N σ 4 3
(9)

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