Statistics How To

Weighting Factor, Statistical Weight and Weight Functions: Definition, Uses

Statistics Definitions > Weighting Factor

Contents:

  1. Weight and the Weighting Factor.
  2. Use in Sampling.
  3. Use in Nuclear Medicine.
  4. Weight Function

1. Weight and the Weighting Factor.

A statistical weight is an amount given to increase or decreased the importance of an item. For example, weights are commonly given for tests and exams in class. For example, a final exam might count for double the points (double the “weight”) of an in-class test.

A weighting factor is a weight given to a data point to assign it a lighter, or heavier, importance in a group. It is usually used for calculating a weighted mean, to give less (or more) importance to group members. It is also used in statistical sampling for adjusting samples and in nuclear medicine for calculating effective doses.

Simple Example

For example, let’s say you take three tests in class. The last test is much harder than the first two tests, so your professor gives it less weight. The weights for the three tests are:

  • Test 1: 40 % of your grade.
  • Test 2: 40 % of your grade.
  • Test 3: 20 % of your grade.

weighting factor

The weighted mean formula uses a weighting factor, wi.

The percents listed after the tests are the weighting factors. For example, test 1 has a weighting factor of 40% while test 3 has a weighting factor of 20%. Let’s say you score 80, 80, and 85 points. The weighted mean for the three tests is found by:

  1. Multiplying your scores by the percentage weights:
    .4(80) = 32
    .4(80) = 32
    .2(95) = 19
  2. Adding the numbers up. 32 + 32 + 19 = 83.

See more examples in Weighted Mean.

2. Use In Sampling

Weighting factors are used in sampling to make samples match the population. For example, let’s say you took a sample of the population and had 41% female and 59% male. You know from census data that females should make up 51% of the population and males 49%. In order to make sure that you have a representative sample, you could add a little more “weight” to data from females. To calculate how much weight you need, divide the known population percentage by the percent in the sample. For this example:

Known population females (51) / Sample Females (41) = 51/41 = 1.24.
Known population males (49) / Sample males (59) = 49/59 = .83.

3. Use In Nuclear Medicine

Weighting factors are used extensively in radiologic and nuclear medicine to calculative effective doses for procedures. The calculations for Tissue Weighting Factors (sometimes called Radiologic Weighting Factors) account for the fact that different parts of the body absorb radiation at different rates.

A tissue weighting factor(WT) is assigned to body parts, with more radiosensitive parts given higher weighting factors.
Effective dose = individual organ dose values * WT.

Tissue weighting factors (ICRP) are:

  • WT = 0.12: stomach, colon, lung, red bone marrow, breast, remainder tissues,
  • WT = 0.08: gonads,
  • WT = 0.04: urinary bladder, oesophagus, liver, thyroid,
  • WT = 0.01: bone surface, skin, brain, salivary glands.

4. What is a Weight Function?

You may want to read this first: What is a Function?

Weights give some elements of a set more weight than others; The weight function is a formula used to apply weights to data. It is used to compensate for known bias, or to give extra influence to data points which matter more for whatever reason. It’s used in for a wide variety of applications from adjusting for outliers to calculating integrals.

A weight function produces a weighted mean or a weighted sum.

Basic Weight Functions: Discrete Weights

Weight functions can be used for either discrete variables or continuous variables. When we are looking at a discrete data set, we define our weight function w as a positive, countable function over the set of interest, which we’ll call A. If we want all our elements to have equal weight, i.e., we want our calculations to be unweighted, the weight function becomes simply w(a) := 1.

If f is any real valued function on f, we know that we can calculate the unweighted sum of f on A with the basic formula




But when, for whatever reason, we need to weight our data with a weight function w:A -> R+

our weighted sum becomes
weighted sum

Suppose A was nonempty and finite. Then we know our unweighted mean (average) is given by


The weighted mean, or weighted average, would be given by


Reasons Behind Weight Functions

There are a number of reasons why a weight function may be used in statistics or engineering applications. This includes:

  • Weighting for accuracy: giving greater weight to more accurate measurements and less weight to measurements known to be less precise.
  • Compensating for bias: giving greater weight to measurements known to be less biased.
  • Accounting for significance: in engineering applications, a weighting function may simply reflect the relative influence of various forces being applied.

References:
European Nuclear Society. Tissue Weighting Factor. Retrieved 9/20/2006 from: https://www.euronuclear.org/info/encyclopedia/t/tissue-weight-factor.htm
International Commission on Radiological Protection. The 2007 Recommendations of the International Commission on Radiological Protection. ICRP publication 103. Ann ICRP. 2007;37 (2-4): 1-33. Available from PubMed.

Grochenig, Karlheinz. Weight Functions in Time-Frequency Analysis. Retrieved from http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.853.2469&rep=rep1&type=pdf on May 17, 2018.

NIST Engineering Statistics Handbook. 4.6.3.4 Weighting to Improve Fit. Retrieved from https://www.itl.nist.gov/div898/handbook/pmd/section6/pmd634.htm on May 19, 2018.

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