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Handout: Descriptive Statistics - Numerical Methods → Fundamental concepts: Population parameter; Point estimate; Sample statistic; Sampling error

Concept Definition Symbol Excel equivalent

or formula

Number of observations

Population size / sample size / number of entries in a data set

n COUNT()

Data set A data set / population / sample of observations or measurements

x1, x2, …, xn A range of cells, e.g. (A2:A52)

Minimum Smallest measurement MIN()

Maximum Largest measurement MAX()

Summation/ Aggregate

All measurements added up ∑ 𝒙𝒊

𝒏

𝒊=𝟏 SUM()

Population mean The sum (aggregate) divided by the number of measurements; the average

μ (‘mew’) AVERAGE()

Sample mean The sum (aggregate) divided by the number of measurements; the average

x̄ (x-bar) AVERAGE()

Median The middlemost measurement Md MEDIAN()

Mode The measurement that occurs most frequently Mo MODE()

Range Difference between Maximum and Minimum MAX()-MIN()

Population Variance Average of the squared deviations from the mean σ2 (sigma square)

(also VAR) VAR.P()

Sample Variance Sum of squared deviations from the mean, divided by (n-1)

s2

(also VAR) VAR.S()

Population Standard Deviation

Square root of the population variance σ (sigma) (also SD)

STDEV.P()

Sample Standard Deviation

Square root of the sample variance s (also SD) STDEV.S()

Coefficient of Variation

A measure of the variation of a sample or population relative to its mean

SD/mean

z-score Deviation of a data value from the mean, measured in standard deviations

z=(x-mean)/SD

Tolerance Interval An interval of several SDs around the population mean that contains a specified percentage of a population (according to the Empirical Rule)

Standard Error of the Sample Mean

A measure of the accuracy with which the sample mean estimates the population mean.

SE=SD/√n

Confidence

Interval

An interval of about 2 standard errors around the sample mean, which with a high confidence contains the population mean

Percentile The pth percentile is a value such that p percent of measurements fall at or below that value.

PERCENTILE.EXC()

Quartiles The 25

th, 50

th (i. e. median) and 75

th percentile of

a data set; the three points that divide the data set into four equal groups

Q1, Q2=Md, Q3

Interquartile range Difference between the third and first quartile IQR Q3- Q1

5-number summary Summary of a dataset as: Minimum, First quartile, Median, Third quartile, Maximum

Quartile, Quintile, Decile

A data set ordered from smallest to largest can be thought of as divided into 4, 5 or 10 equal-sized subsets. The terms quartile, quintile or decile may refer either to these subsets (sociology, economics) or to the values that divide them (descriptive statistics, cf. Quartiles).