Formula sheet

$\mu$

population mean

$x$

each observation in the population

$N$

population size

you can observe every member of the population and want its average.

$\bar{x}$

sample mean

$x$

each observation in the sample

$n$

sample size

you have a sample and need its arithmetic average.

$\hat{p}$

sample proportion

$x$

number of observations with the attribute

$n$

sample size

you need the fraction of a sample that has some attribute (such as consumers who would buy a product).

$\mu$

population mean

$x$

each value in the population

$N$

population size

every member of the population is observed.

$\bar{x}$

sample mean

$x$

each value in the sample

$n$

sample size

you summarize a sample. The mean uses every observation, so a single extreme value can move it a lot.

$n$

number of observations

$the data must be ordered firs$

the data must be ordered first

you want to locate the middle of ordered data. For even n, average the two middle values.

$Maximum$

largest value

$Minimum$

smallest value

you need a quick, crude measure of total spread. It uses only two observations, so it ignores everything in between.

$\sigma^{2}$

population variance

$x$

each observation

$\mu$

population mean

$N$

population size

you have all population values and want their variability.

$s^{2}$

sample variance

$x$

each observation

$\bar{x}$

sample mean

$n$

sample size

you have a sample and want its variability.

$\sigma$

population standard deviation

$\sigma^{2}$

population variance

you want spread measured in the original units.

$s$

sample standard deviation

$s^{2}$

sample variance

you want spread in the original units, comparable with the mean.

$CV$

coefficient of variation

$s$

sample standard deviation

$\bar{x}$

sample mean

you compare variability of variables measured in different units or with very different means; the result is a percentage.

$f$

class frequency

$n$

total number of observations

you want the share of the total that falls in a class rather than the raw count.

$relative frequency$

class share of the total, as a fraction

$360^\circ$

the full circle

you are drawing a pie chart and need the central angle of one class's slice.

$k$

number of classes

$n$

number of observations

you must choose how many classes a frequency distribution should have; pick the smallest k that satisfies the rule.

$i$

class interval

$Maximum$

largest data value

$Minimum$

smallest data value

$k$

number of classes

the number of classes is chosen and you need the width of each class.

$L$

lower limit of the class

$U$

upper limit of the same class

you need one typical value to represent a class, for instance to plot a frequency polygon.

$cumulative frequency$

number of observations below a class upper limit

$n$

total number of observations

you want the share of observations below a value rather than a single class share.

$\mu$

population mean

$x$

each observation

$N$

number of observations in the population

you have data for the entire population and want its arithmetic average.

$\bar{x}$

sample mean

$x$

each observation

$n$

number of observations in the sample

you have a sample and want its arithmetic average.

$\bar{x}_w$

weighted mean

$x$

each value

$w$

the weight of that value

values repeat or are not equally important.

$GM$

geometric mean

$x$

each positive value

$n$

number of values

you average growth factors, ratios, or percentages instead of raw amounts.

$GM$

average compound growth rate

$n$

number of periods

$end and start$

values at the two dates

you want the average annual growth rate between two points in time.

$Maximum$

largest value

$Minimum$

smallest value

you want a quick, easily understood measure of the total spread.

$\sigma^2$

population variance

$x$

each observation

$\mu$

population mean

$N$

population size

the data are the whole population.

$\sigma$

population standard deviation

$x$

each observation

$\mu$

population mean

$N$

population size

you want dispersion in the original units.

$s^2$

sample variance

$x$

each observation

$\bar{x}$

sample mean

$n$

sample size

the data are a sample.

$s$

sample standard deviation

$x$

each observation

$\bar{x}$

sample mean

$n$

sample size

you want the sample spread in the original units.

$\bar{x}$

grouped sample mean

$f$

class frequency

$M$

class midpoint

$n$

total number of observations

you only have a frequency distribution.

$s$

grouped sample standard deviation

$f$

class frequency

$M$

class midpoint

$\bar{x}$

grouped mean

$n$

total number of observations

you estimate dispersion from a frequency distribution.

$L_P$

location (position number) of the Pth percentile in the ordered array

$n$

number of observations

$P$

the percentile wanted (25 for Q_1, 50 for the median, 75 for Q_3, 20 for the 2nd decile)

you need to find the position of a quartile, decile or percentile in data sorted from smallest to largest.

$IQR$

interquartile range

$Q_1$

first quartile

$Q_3$

third quartile

you want the spread of the middle 50% of the data, the width of the box.

$L$

lower limit below which a value is an outlier

$U$

upper limit above which a value is an outlier

$IQR$

interquartile range

you must decide whether a value is an outlier: any observation outside these limits is flagged (an asterisk in a box plot).

$sk$

coefficient of skewness

$\bar{x}$

sample mean

$Median$

sample median

$s$

sample standard deviation

you want a quick numerical measure of the direction and degree of skewness from the mean, median and standard deviation.

$sk$

coefficient of skewness

$n$

number of observations

$x$

each observation

$\bar{x}$

sample mean

$s$

sample standard deviation

you are reading software output or asked for the software method; it is built from the cubed deviations of each value from the mean in standard-deviation units.

$\bar{x}$

sample mean

$\sum x$

sum of all observations

$n$

number of observations

you need the arithmetic average of a sample.

$s$

sample standard deviation

$x$

each observation

$\bar{x}$

sample mean

$n$

number of observations

you need the typical distance of observations from the sample mean.