science

Partition of sums of squares

Partition of sums of squares is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Partition of sums of squares rather than just read about it. In short: The partition of sums of squares is a concept that permeates much of inferential statistics and descriptive statistics. More properly, it is the partitioning of sums of squared deviations or errors.

Key takeaways

Partition of sums of squares belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Partition of sums of squares to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Partition of sums of squares from memory before moving on to harder problems.

Reference excerpt

The partition of sums of squares is a concept that permeates much of inferential statistics and descriptive statistics. More properly, it is the partitioning of sums of squared deviations or errors. Mathematically, the sum of squared deviations is an unscaled, or unadjusted measure of dispersion (also called variability). When scaled for the number of degrees of freedom, it estimates the variance, or spread of the observations about their mean value. Partitioning of the sum of squared deviations into various components allows the overall variability in a dataset to be ascribed to different types or sources of variability, with the relative importance of each being quantified by the size of each component of the overall sum of squares.

Background The distance from any point in a collection of data, to the mean of the data, is the deviation. This can be written as y i − y ¯ {\displaystyle y_{i}-{\overline {y}}} , where y i {\displaystyle y_{i}} is the ith data point, and y ¯ {\displaystyle {\overline {y}}} is the estimate of the mean. If all such deviations are squared, then summed, as in ∑ i = 1 n ( y i − y ¯ ) 2 {\displaystyle \sum _{i=1}^{n}\left(y_{i}-{\overline {y}}\,\right)^{2}} , this gives the "sum of squares" for these data. When more data are added to the collection the sum of squares will increase, except in unlikely cases such as the new data being equal to the mean. So usually, the sum of squares will grow with the size of the data collection. That is a manifestation of the fact that it is unscaled. In many cases, the number of degrees of freedom is simply the number of data points in the collection, minus one. We write this as n − 1, where n is the number of data points. Scaling (also known as normalizing) means adjusting the sum of squares so that it does not grow as the size of the data collection grows. This is important when we want to compare samples of different sizes, such as a sample of 100 people compared to a sample of 20 people. If the sum of squares were not normalized, its value would always be larger for the sample of 100 people than for the sample of 20 people. To scale the sum of squares, we divide it by the degrees of freedom, i.e., calculate the sum of squares per degree of freedom, or variance. Standard deviation, in turn, is the square root of the variance. The above describes how the sum of squares is used in descriptive statistics; see the article on total sum of squares for an application of this broad principle to inferential statistics.

Partitioning the sum of squares in linear regression Theorem. Given a linear regression model y i = β 0 + β 1 x i 1 + ⋯ + β p x i p + ε i {\displaystyle y_{i}=\beta _{0}+\beta _{1}x_{i1}+\cdots +\beta _{p}x_{ip}+\varepsilon _{i}} including a constant β 0 {\displaystyle \beta _{0}} , based on a sample ( y i , x i 1 , … , x i p ) , i = 1 , … , n {\displaystyle (y_{i},x_{i1},\ldots ,x_{ip}),\,i=1,\ldots ,n} containing n observations, the total sum of squares T S S = ∑ i = 1 n ( y i − y ¯ ) 2 {\displaystyle \mathrm {TSS} =\sum _{i=1}^{n}(y_{i}-{\bar {y}})^{2}} can be partitioned as follows into the explained sum of squares (ESS) and the residual sum of squares (RSS):

T S S = E S S + R S S , {\displaystyle \mathrm {TSS} =\mathrm {ESS} +\mathrm {RSS} ,}

where this equation is equivalent to each of the following forms:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Partition of sums of squares

Start with the simplest possible case. Write down what Partition of sums of squares claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Partition of sums of squares before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Partition of sums of squares ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Partition of sums of squares

In research: Partition of sums of squares appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Partition of sums of squares in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Partition of sums of squares is common in secondary-school and first-year university syllabi. It links to neighbouring topics Analysis of variance, Least squares, so understanding it makes those chapters shorter.
In everyday life: Look for Partition of sums of squares outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Partition of sums of squares in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Partition of sums of squares means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Partition of sums of squares out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Partition of sums of squares in simple terms?

The partition of sums of squares is a concept that permeates much of inferential statistics and descriptive statistics. More properly, it is the partitioning of sums of squared deviations or errors.

Why does Partition of sums of squares matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Partition of sums of squares?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Partition of sums of squares.