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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%.

#### 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:

Q1Q22 Q1 Q2 2
(4)

#### Variance and Standard Deviation

The variance in a population is:

σ2= X XμN 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 XMN 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 XMN1 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 Xμ3Nσ3 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 Xμ4Nσ43 Kurtosis X X μ 4 N σ 4 3
(9)

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