What symbols are used to represent the mean and standard deviation of the sampling distribution of?

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Here are symbols for various sample statistics and the corresponding population parameters. They are not repeated in the list below.

μ and σ can take subscripts to show what you are taking the mean or standard deviation of. For instance, σx̅ (“sigma sub x-bar”) is the standard deviation of sample means, or standard error of the mean.

• b

  = y intercept of a line. Defined here in Chapter 4. (Some statistics books use b0.)

• BD

  or

  BPD

  = binomial probability distribution. Defined here in Chapter 6.

• CI

  = confidence interval. Defined here in Chapter 9.

• CLT

  = Central Limit Theorem. Defined here in Chapter 8.

• d

  = difference between paired data. Defined here in Chapter 11.

• df

  or

  ν

  “nu” = degrees of freedom in a Student’s t or χ² distribution. Defined here in Chapter 9. Defined here in Chapter 12.

• DPD

  = discrete probability distribution. Defined here in Chapter 6.

• E

  = margin of error, a/k/a maximum error of the estimate. Defined here in Chapter 9.

• f

  = frequency. Defined here in Chapter 2.

• f/n

  = relative frequency. Defined here in Chapter 2.

• HT

  = hypothesis test. Defined here in Chapter 10.

• Ho

  = null hypothesis. Defined here in Chapter 10.

• H1

  or

  Ha

  = alternative hypothesis. Defined here in Chapter 10.

• IQR

  = interquartile range, Q3−Q1. Defined here in Chapter 3.

• m

  = slope of a line. Defined here in Chapter 4. (The TI-83 uses a and some statistics books use b1.)

• M

  or Med = median of a sample. Defined here in Chapter 3.

• n

  = sample size, number of data points. Defined here in Chapter 2. Also, number of trials in a probability experiment with a binomial model. Defined here in Chapter 6.

• N

  = population size.

• ND

  = normal distribution, whose graph is a bell-shaped curve; also “normally distributed”. Defined here in Chapter 7.

• p

  = probability value. The specific meaning depends on context.

  In geometric and binomial probability distributions, p is the probability of “success” (defined here in Chapter 6) on any one trial and q = (1−p) is the probability of “failure” (the only other possibility) on any one trial.

  In hypothesis testing, p is the calculated p-value (defined here in Chapter 10), the probability that rejecting the null hypothesis would be a wrong decision.

  In tests of population proportions, p stands for population proportion and p̂ for sample proportion (see table above).

• P(A)

  = the probability of event A.

• P(AC)

  or

  P(not A)

  = the probability that A does not happen. Defined here in Chapter 5.

• P(B | A)

  = the probability that event B will happen, given that event A definitely happens. It’s usually read as the probability of B given A. Defined here in Chapter 5.

  Caution! The order of A and B may seem backward to you at first.

• P80

  or

  P80

  = 80th percentile (

  Pk

  or

  Pk

  = k-th percentile) Defined here in Chapter 3.

• q

  = probability of failure on any one trial in binomial or geometric distribution, equal to (1−p) where p is the probability of success on any one trial. Defined here in Chapter 6.

• Q1

  or

  Q1

  = first quartile (

  Q3

  or

  Q3

  = third quartile) Defined here in Chapter 3.

• r

  = linear correlation coefficient of a sample. Defined here in Chapter 4.

• R²

  = coefficient of determination. Defined here in Chapter 4.

• s

  = standard deviation of a sample. Defined here in Chapter 3.

• SD

  (or s.d.) = standard deviation. Defined here in Chapter 3.

• SEM

  = standard error of the mean (symbol is σ

  x̅

  ). Defined here in Chapter 8.

• SEP

  = standard error of the proportion (symbol is σ

  p̂

  ). Defined here in Chapter 8.

• X

  (capital X) = a variable.

• x

  (lower-case x) = one data value (“raw score”). As a column heading, x means a series of data values.

• x̅

  “x-bar” = mean of a sample. Defined here in Chapter 3.

• x̃

  “x-tilde” = median of a sample. Defined here in Chapter 3.

• ŷ

  “y-hat” = predicted average y value for a given x, found by using the regression equation. Defined here in Chapter 4.

• z

  = standard score or z-score. Defined here in Chapter 3.

• z(area)

  or

  zarea

  = the z-score, such that that much of the area under the normal curve lies to the right of that z. This is not a multiplication! (See The z Function.)

• 5 Nov 2020: Convert document to HTML5, and italicize the variables.
• 14 Feb 2018: Add

  x̃

  for the median, as suggested by reader “Trone”.
• (intervening changes suppressed)
• 27 Sept 2002: New article.

This web page describes how symbols are used on the Stat Trek web site to represent numbers, variables, parameters, statistics, etc.

Capitalization

In general, capital letters refer to population attributes (i.e., parameters); and lower-case letters refer to sample attributes (i.e., statistics). For example,

• P refers to a population proportion; and p, to a sample proportion.
• X refers to a set of population elements; and x, to a set of sample elements.
• N refers to population size; and n, to sample size.

Greek vs. Roman Letters

Like capital letters, Greek letters refer to population attributes. Their sample counterparts, however, are usually Roman letters. For example,

• μ refers to a population mean; and x, to a sample mean.
• σ refers to the standard deviation of a population; and s, to the standard deviation of a sample.

Population Parameters

By convention, specific symbols represent certain population parameters. For example,

• μ refers to a population mean.
• σ refers to the standard deviation of a population.
• σ2 refers to the variance of a population.
• P refers to the proportion of population elements that have a particular attribute.
• Q refers to the proportion of population elements that do not have a particular attribute, so Q = 1 - P.
• ρ is the population correlation coefficient, based on all of the elements from a population.
• N is the number of elements in a population.

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Sample Statistics

By convention, specific symbols represent certain sample statistics. For example,

• x refers to a sample mean.
• s refers to the standard deviation of a sample.
• s2 refers to the variance of a sample.
• p refers to the proportion of sample elements that have a particular attribute.
• q refers to the proportion of sample elements that do not have a particular attribute, so q = 1 - p.
• r is the sample correlation coefficient, based on all of the elements from a sample.
• n is the number of elements in a sample.

Simple Linear Regression

• Β0 is the intercept constant in a population regression line.
• Β1 is the regression coefficient (i.e., slope) in a population regression line.
• R2 refers to the coefficient of determination.
• b0 is the intercept constant in a sample regression line.
• b1 refers to the regression coefficient in a sample regression line (i.e., the slope).
• sb1 refers to the refers to the standard error of the slope of a regression line.

Probability

• n! refers to the factorial value of n.
• nPr refers to the number of permutations of n things taken r at a time.
• nCr refers to the number of combinations of n things taken r at a time.

• Z or z refers to a standardized score, also known as a z-score.
• zα refers to the standardized score that has a cumulative probability equal to 1 - α.
• tα refers to the t statistic that has a cumulative probability equal to 1 - α.
• fα refers to a f statistic that has a cumulative probability equal to 1 - α.
• fα(v1, v2) is a f statistic with a cumulative probability of 1 - α, and v1 and v2 degrees of freedom.
• Χ2 refers to a chi-square statistic.

Throughout the site, certain symbols have special meanings. For example,

• Σ is the summation symbol, used to compute sums over a range of values.
• Σx or Σxi refers to the sum of a set of n observations. Thus, Σxi = Σx = x1 + x2 + . . . + xn.
• sqrt refers to the square root function. Thus, sqrt(4) = 2 and sqrt(25) = 5.
• Var(X) refers to the variance of the random variable X.
• SD(X) refers to the standard deviation of the random variable X.
• SE refers to the standard error of a statistic.
• ME refers to the margin of error.
• DF refers to the degrees of freedom.

If you would like to cite this web page, you can use the following text:

Berman H.B., "Stat Trek Statistics Notiation", [online] Available at: https://stattrek.com/statistics/notation URL [Accessed Date: 8/10/2022].

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