Key Statistical Measures: Location and Variation in Data

Sample Characteristics: Measures of Location and Variation

Measures of Location

• Average

  The average is a central value, often referred to as a measure of central tendency.

• Mean

  • Also known as the arithmetic mean.
  • Found by dividing the sum of all values by the total number of values.
  • Formula: Mean = (Sum of Quantities) / (Number of Quantities)
  • Example: Find the mean of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
    • Sum of values: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55
    • Number of values = 10
    • Mean of values = 55 / 10 = 5.5

• Median

  The median of a set is the middle number when the values are arranged in order from lowest to highest.

• Mode

  • The mode is the value that occurs most frequently in a set of values, also known as the modal value.
  • Example: Find the mode of 1, 2, 2, 3, 4, 4, 5, 5, 5, 5, 7, 8, 8, and 9.
  • The modal value is 5.

Measures of Variation

• Range

  The range is the difference between the highest and the lowest values in a dataset.

• Variance

  Variance is a measure of how far a set of numbers are spread out from each other.

  It is calculated as the sum of squared deviations of each individual value from the mean, divided by n-1, where n is the sample size.

• Standard Deviation (σ)

  Standard Deviation (σ) measures how spread out the data is.

  It is the square root of the variance.

  How to Calculate Standard Deviation:

  1. Calculate the mean for the data set.
  2. Calculate the deviation by subtracting the mean from each value.
  3. Square each individual deviation.
  4. Add up the squared deviations.
  5. Divide by one less than the sample size (n-1).
  6. Take the square root of the result.

• Coefficient of Variation

  The Coefficient of Variation is a normalized measure of the dispersion of a probability distribution.

• Standard Error

  Standard Error is the standard deviation of the sampling distribution of a statistic.

• Percentiles (P)

  Percentiles (P) indicate the value below which a given percentage of observations in a group of observations falls.

  For example, the 50^th percentile is equal to the median.