average

(noun)

any measure of central tendency, especially any mean, the median, or the mode

Related Terms

Examples of average in the following topics:

  • Expected Values of Discrete Random Variables

    • The expected value of a random variable is the weighted average of all possible values that this random variable can take on.
    • The weights used in computing this average are probabilities in the case of a discrete random variable.
    • If all outcomes $x_i$ are equally likely (that is, $p_1 = p_2 = \dots = p_i$), then the weighted average turns into the simple average.
    • This is intuitive: the expected value of a random variable is the average of all values it can take; thus the expected value is what one expects to happen on average.
    • If the outcomes $x_i$ are not equally probable, then the simple average must be replaced with the weighted average, which takes into account the fact that some outcomes are more likely than the others.

  • A Graph of Averages

    • A graph of averages and the least-square regression line are both good ways to summarize the data in a scatterplot.
    • Sometimes, a graph of averages is used to show a pattern between the $y$ and $x$ variables.
    • In a graph of averages, the $x$-axis is divided up into intervals.
    • The averages of the $y$ values in those intervals are plotted against the midpoints of the intervals.
    • The points on a graph of averages do not usually line up in a straight line, making it different from the least-squares regression line.

  • Homework

    • Let $\bar{X}$ be the random variable of averages.
    • (Use a random number generator. ) Continue averaging 5 pieces together until you have 10 averages.
    • List those 10 averages.
    • (Use a random number generator. ) Continue averaging 5 pieces together until you have 15 averages.
    • List those 15 averages.

  • Difference of two means exercises

    • The average and standard deviation in 2004 were 257 and 39, respectively.
    • In 2008, the average and standard deviation were 260 and 38, respectively.
    • We are 90% confident that the average score in 2008 was 0.16 to 5.84 points higher than the average score in 2004.
    • 5.9 (a) H0 : µ2008 = µ2004 → µ2004−2008 = 0 (Average math score in 2008 is equal to average math score in 2004. ) HA : µ2008 / 6= µ2004 → µ2004 −2008 / 6= 0 (Average math score in 2008 is different than average math score in 2004. ) Conditions necessary for inference were checked in Exercise 5.7.
    • We are 95% confident that the average body fat percentage in men is 11.32% to 10.88% lower than the average body fat percentage in women.

  • What Does the Law of Averages Say?

    • The law of averages says it's due to land on black!
    • Some people interchange the law of averages with the law of large numbers, but they are different.
    • The law of averages is not a mathematical principle, whereas the law of large numbers is.
    • As the number of rolls in this run increases, the average of the values of all the results approaches 3.5.
    • Evaluate the law of averages and distinguish it from the law of large numbers.

  • The Regression Method

    • The regression method utilizes the average from known data to make predictions about new data.
    • The average SAT score is 560, with a standard deviation of 75.
    • The average first year GPA is 2.8, with a standard deviation of 0.5.
    • With no other information given, it is best to predict using the average.
    • Instead of just predicting 2.8, this time we look at the graph of averages and predict her GPA is whatever the average is of all the students in our sample who also scored a 680 on the SAT.

  • Mean: The Average

    • In the simplest cases, the measure of central tendency is an average of a set of measurements, the word average being variously construed as mean, median, or other measure of location, depending on the context.
    • An average is a measure of the "middle" or "typical" value of a data set.
    • If the numbers are not the same, the average is calculated by combining the numbers from the list in a specific way and computing a single number as being the average of the list.
    • When we think of means, or averages, we are typically thinking of the arithmetic mean.
    • Define the average and distinguish between arithmetic, geometric, and harmonic means.

  • The Root-Mean-Square

    • Computing the average of this set of numbers wouldn't tell us much because the negative numbers cancel out the positive numbers, resulting in an average of zero.
    • This gives us the "middle value" but not a sense of the average magnitude.
    • One possible method of assigning an average to this set would be to simply erase all of the negative signs.
    • This would lead us to compute an average of 5.6.
    • The root-mean-square is always greater than or equal to the average of the unsigned values.

  • The Sample Average

    • The sample average/mean can be calculated taking the sum of every piece of data and dividing that sum by the total number of data points.
    • The sample average (also called the sample mean) is often referred to as the arithmetic mean of a sample, or simply, $\bar{x}$ (pronounced "x bar").
    • The arithmetic mean is the "standard" average, often simply called the "mean".
    • The mean is the arithmetic average of a set of values, or distribution; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely (mode).

  • Interpreting Significant Results

    • If the sample mean for the obese patients is significantly lower than the sample mean for the average-weight patients, then one should conclude that the population mean for the obese patients is lower than the sample mean for the average-weight patients.
    • If the former of these is rejected, then the conclusion is that the population mean for obese patients is lower than that for average-weight patients.
    • If the latter is rejected, then the conclusion is that the population mean for obese patients is higher than that for average-weight patients.