Lebesgue measure

(noun)

The unique complete translation-invariant measure for the $\sigma$-algebra which contains all $k$-cells—in and which assigns a measure to each $k$-cell equal to that $k$-cell's volume (as defined in Euclidean geometry: i.e., the volume of the $k$-cell equals the product of the lengths of its sides).

Examples of Lebesgue measure in the following topics:

  • Continuous Probability Distributions

    • Mathematicians also call such a distribution "absolutely continuous," since its cumulative distribution function is absolutely continuous with respect to the Lebesgue measure $\lambda$.
    • For example, if one measures the width of an oak leaf, the result of 3.5 cm is possible; however, it has probability zero because there are uncountably many other potential values even between 3 cm and 4 cm.

  • Collecting and Measuring Data

    • There are four main levels of measurement: nominal, ordinal, interval, and ratio.
    • Nominal measurements have no meaningful rank order among values.
    • Interval measurements have meaningful distances between measurements defined, but the zero value is arbitrary (as in the case with longitude and temperature measurements in Celsius or Fahrenheit).
    • Measurement processes that generate statistical data are also subject to error.
    • Distinguish between the nominal, ordinal, interval and ratio methods of data measurement.

  • Student Learning Outcomes

    • Recognize, describe, and calculate the measures of location of data: quartiles and percentiles.
    • Recognize, describe, and calculate the measures of the center of data: mean, median, and mode.
    • Recognize, describe, and calculate the measures of the spread of data: variance, standard deviation, and range.

  • Averages of Qualitative and Ranked Data

    • In statistics, levels of measurement, or scales of measure, are types of data that arise in the theory of scale types developed by the psychologist Stanley Smith Stevens.
    • The mode, i.e. the most common item, is allowed as the measure of central tendency for the nominal type.
    • Examples include, on one hand, dichotomous data with dichotomous (or dichotomized) values such as "sick" versus "healthy" when measuring health, "guilty" versus "innocent" when making judgments in courts, or "wrong/false" versus "right/true" when measuring truth value.
    • The median, i.e. middle-ranked, item is allowed as the measure of central tendency; however, the mean (or average) as the measure of central tendency is not allowed.
    • Categorize levels of measurement and identify the appropriate measures of central tendency.

  • Comparing Measures of Central Tendency

    • How do the various measures of central tendency compare with each other?
    • Differences among the measures occur with skewed distributions.
    • Measures of central tendency are shown in Table 1.
    • The geometric mean is lower than all measures except the mode.
    • Table 2 shows the measures of central tendency for these data.

  • Chance Error

    • Every measurement has an inherent uncertainty.
    • We therefore need to give some indication of the reliability of measurements and the uncertainties of the results calculated from these measurements.
    • They cannot be avoided; they are part of the measuring process.
    • Uncertainties are measures of random errors.
    • Three measurements of a single object might read something like 0.9111g, 0.9110g, and 0.9112g.

  • Range

    • The range is a measure of the total spread of values in a quantitative dataset.
    • In statistics, the range is a measure of the total spread of values in a quantitative dataset.
    • Unlike other more popular measures of dispersion, the range actually measures total dispersion (between the smallest and largest values) rather than relative dispersion around a measure of central tendency.
    • The range is measured in the same units as the variable of reference and, thus, has a direct interpretation as such.
    • The mid-range is the midpoint of the range; as such, it is a measure of central tendency.

  • Repeated Measures Design

    • Repeated measures analysis of variance (rANOVA) is one of the most commonly used statistical approaches to repeated measures designs.
    • Other studies compare the same measure under two or more different conditions.
    • Repeated measures analysis of variance (rANOVA) is one of the most commonly used statistical approaches to repeated measures designs.
    • While there are many advantages to repeated-measures design, the repeated measures ANOVA is not always the best statistical analyses to conduct.
    • Evaluate the significance of repeated measures design given its advantages and disadvantages

  • Bias

    • Every measurement has an inherent uncertainty.
    • We therefore need to give some indication of the reliability of measurements and the uncertainties of the results calculated from these measurements.
    • Systematic errors are biases in measurement which lead to a situation wherein the mean of many separate measurements differs significantly from the actual value of the measured attribute.
    • Accuracy (or validity) is a measure of the systematic error.
    • Accuracy is a measure of how well an experiment measures what it was trying to measure.

  • Median

    • A measure of central tendency (also referred to as measures of center or central location) is a summary measure that attempts to describe a whole set of data with a single value that represents the middle or center of its distribution.
    • There are three main measures of central tendency: the mode, the median and the mean .
    • Each of these measures describes a different indication of the typical or central value in the distribution.
    • The median is less affected by outliers and skewed data than the mean, and is usually the preferred measure of central tendency when the distribution is not symmetrical.
    • Identify the median in a data set and distinguish it's properties from other measures of central tendency.