<p><strong>IQ scores in general:</strong> An IQ score is a normative score. The norm group is typically defined as the general population, and where the respondent is a child, the norm group is defined in terms of the general population of children of that same age. IQ scores typically have a mean of 100 and a standard deviation of 15.</p>

<p>In order to get an estimate of performance on the test of a norm group, it is necessary to get a sample from that population. Large and unbiased samples from the population are preferred. </p>

<p>Before discussing how to get IQ scores for children, let's focus on adults where there is no age adjustment. Assume we get a sample of 1,000 adult participants that is hopefully representative of a given adult population and we administer the intelligence test to the participants. We score the test and we then get a set of raw test scores. The most common scoring system is one point per correct question, but many other systems are possible including penalising for incorrect answers, weighting some questions more than others, and so on. Once you have raw scores, there are now several options for creating a norm table that converts raw scores into IQ scores. The simplest approach is to get the raw mean and standard deviation, convert to <a href="http://en.wikipedia.org/wiki/Standard_score" rel="noreferrer">z-scores</a>, and then convert to IQ units:</p>

<p>$$\begin{array}{c} {z_i}{\rm{ \;=\; }}\frac{{{\rm{x}}_i}{\rm{ \;-\; }}{{\rm{MEAN}}}}{{\rm{SD}}}\\ {\rm{IQ}_i}{\rm{ \;= 100 }} +{\rm{ }}{z_i}{} \times {\rm{ 15}} \end{array}$$</p>

<p>Where MEAN is the mean of the normative sample, SD is the SD of the normative sample, $x_i$ is the the score of respondent $i$, $z_i$ is respondent $i$'s z-score, and $\rm{IQ}_i$ is participant $i$'s IQ score.</p>

<p>Angoff (1984, the <a href="http://www.ets.org/Media/Research/pdf/Angoff.Scales.Norms.Equiv.Scores.pdf" rel="noreferrer">text is available online for free from ETS</a>) calls this the linear transformation (standard scores) approach. There are more advanced methods which you can read about in Angoff (1984) such as percentile-derived linear scale, percentile rank scale, normalised scale. </p>

<p><strong>Age adjustments:</strong> Angoff (1984) also discusses how to construct age equivalent norms.</p>

<p>In summary, if you want to get IQ scores for children, then your aim is to estimate the distribution of raw test scores in the general population for each age group of interest. There are various ways of doing this. The simplest approach is to get large samples of children at each year age of interest, and get means and standard deviations, and use them as above as appropriate. A more sophisticated approach involves smoothing the data to get an estimate of the systematic changes in both the mean and the standard deviation with age. Such a smoothed model can also be applied to get more fine grained estimates of a child who is say not exactly 10 years old, but rather 10 years and 5 months. There are many variations on this approach, and there are many issues to consider regarding the applicability of the same intelligence test to adults and children, especially as you get towards early childhood. </p>

<p>See also this earlier question about <a href="https://cogsci.stackexchange.com/questions/3595/how-iq-is-calculated-by-age?rq=1">Raven's Matrices and age adjusted norms</a>.</p>

<h3>References</h3>

<ul>
<li>Angoff, W. (1984). Scales, norms, and equivalent scores. Educational Measurement: Origins, Theories, and Explications, 2, 121. <a href="http://www.ets.org/Media/Research/pdf/Angoff.Scales.Norms.Equiv.Scores.pdf" rel="noreferrer">PDF</a></li>
</ul>

CC BY-SA 3.0
2013-09-03T07:20:14.453Z
Excellent answer very informative thank you
ProfessorIQ
CC BY-SA 3.0
2015-12-05T03:41:44.593Z
"Before discussing how to get IQ scores for children, let's focus on adults where there is no age adjustment." - Jeromy, what does this statement mean? Is age adjustment not required in Adults, and if yes, why?
Vaidy
CC BY-SA 3.0
2015-12-06T17:26:56.927Z
Age adjustment is not routinely done for adults. raw IQ is fairly stable over early-middle adulthood, whereas raw scores on IQ tests change a lot over childhood.
Jeromy Anglim
CC BY-SA 3.0
2018-03-08T10:15:43.130Z
When you say "smoothing" does it mean something like running an OLS regression of the raw score on age (and maybe age-squared, age-cubed etc), take the residuals, and then center them to have mean 100 and sd 15?
pietro
CC BY-SA 3.0
2018-03-08T22:37:59.247Z
@pietro; Yep. Something like that. Obviously, there's lots of options for the model both in what the mean is and whether and how you let the error variance vary over time. A major benefit is that you can get more granular predictions particularly over childhood where the difference between ages 7 and 0 months and 7 and 8 months would be meaningful.
Jeromy Anglim
CC BY-SA 3.0
2013-09-03T00:22:17.787Z
2013-09-06T11:48:06.173Z
2017-04-13T12:56:16.417Z
Jeromy Anglim
<p>I am having a hard time grasping how IQ scores are calculated. I think it has something to do with their age score ratio multiplied by 100.</p>

<p>Say I have a test that holds 30 questions. How would I generate from that test an IQ score?
Would each question be worth 1 point? Then I would divide by age and multiply by 100?</p>

<p>So if someone got 26 questions correct and they were 18 it would be</p>

<pre><code>26 / 18 = 1.44
1.44 * 100 = 144
</code></pre>

<p>Is this the general way of doing things?</p>

<p>I have researched the internet with no definitive answer to this even on wikipedia.</p>

CC BY-SA 3.0
2013-09-03T11:52:51.543Z
I've added your question about speed and IQ scores separately here: http://cogsci.stackexchange.com/questions/4491/to-what-extent-is-time-to-complete-an-intelligence-test-incorporated-into-the-iq
Jeromy Anglim
<intelligence><developmental-psychology><statistics><test>
How to calculate IQ score based on raw score and adjust for age?
ProfessorIQ
# How to calculate IQ score based on raw score and adjust for age?

(Asked by: username_0 on 2013-09-02)

I am having a hard time grasping how IQ scores are calculated. I think it has something to do with their age score ratio multiplied by 100.
Say I have a test that holds 30 questions. How would I generate from that test an IQ score?
Would each question be worth 1 point? Then I would divide by age and multiply by 100?
So if someone got 26 questions correct and they were 18 it would be
26 / 18 = 1.44
1.44 * 100 = 144

Is this the general way of doing things?
I have researched the internet with no definitive answer to this even on wikipedia.

**Tags:** intelligence, developmental-psychology, statistics, test

> **Comments:**
> **username_1**: I've added your question about speed and IQ scores separately here: http://cogsci.stackexchange.com/questions/4491/to-what-extent-is-time-to-complete-an-intelligence-test-incorporated-into-the-iq  

---

## 1 answers

### Answer by username_1 on 2013-09-03. Score: 11 | Accepted answer

**IQ scores in general:** An IQ score is a normative score. The norm group is typically defined as the general population, and where the respondent is a child, the norm group is defined in terms of the general population of children of that same age. IQ scores typically have a mean of 100 and a standard deviation of 15.

In order to get an estimate of performance on the test of a norm group, it is necessary to get a sample from that population. Large and unbiased samples from the population are preferred.

Before discussing how to get IQ scores for children, let's focus on adults where there is no age adjustment. Assume we get a sample of 1,000 adult participants that is hopefully representative of a given adult population and we administer the intelligence test to the participants. We score the test and we then get a set of raw test scores. The most common scoring system is one point per correct question, but many other systems are possible including penalising for incorrect answers, weighting some questions more than others, and so on. Once you have raw scores, there are now several options for creating a norm table that converts raw scores into IQ scores. The simplest approach is to get the raw mean and standard deviation, convert to [z-scores](http://en.wikipedia.org/wiki/Standard_score), and then convert to IQ units:

$$\begin{array}{c} {z\_i}{\rm{ \;=\; }}\frac{{{\rm{x}}\_i}{\rm{ \;-\; }}{{\rm{MEAN}}}}{{\rm{SD}}}\\ {\rm{IQ}\_i}{\rm{ \;= 100 }} +{\rm{ }}{z\_i}{} \times {\rm{ 15}} \end{array}$$

Where MEAN is the mean of the normative sample, SD is the SD of the normative sample, $x\_i$ is the the score of respondent $i$, $z\_i$ is respondent $i$'s z-score, and $\rm{IQ}\_i$ is participant $i$'s IQ score.

Angoff (1984, the [text is available online for free from ETS](http://www.ets.org/Media/Research/pdf/Angoff.Scales.Norms.Equiv.Scores.pdf)) calls this the linear transformation (standard scores) approach. There are more advanced methods which you can read about in Angoff (1984) such as percentile-derived linear scale, percentile rank scale, normalised scale.

**Age adjustments:** Angoff (1984) also discusses how to construct age equivalent norms.

In summary, if you want to get IQ scores for children, then your aim is to estimate the distribution of raw test scores in the general population for each age group of interest. There are various ways of doing this. The simplest approach is to get large samples of children at each year age of interest, and get means and standard deviations, and use them as above as appropriate. A more sophisticated approach involves smoothing the data to get an estimate of the systematic changes in both the mean and the standard deviation with age. Such a smoothed model can also be applied to get more fine grained estimates of a child who is say not exactly 10 years old, but rather 10 years and 5 months. There are many variations on this approach, and there are many issues to consider regarding the applicability of the same intelligence test to adults and children, especially as you get towards early childhood.

See also this earlier question about [Raven's Matrices and age adjusted norms](https://cogsci.stackexchange.com/questions/3595/how-iq-is-calculated-by-age?rq=1).

### References

* Angoff, W. (1984). Scales, norms, and equivalent scores. Educational Measurement: Origins, Theories, and Explications, 2, 121. [PDF](http://www.ets.org/Media/Research/pdf/Angoff.Scales.Norms.Equiv.Scores.pdf)

> **Comments:**
> **username_0**: Excellent answer very informative thank you  
> **username_2**: "Before discussing how to get IQ scores for children, let's focus on adults where there is no age adjustment." - Jeromy, what does this statement mean? Is age adjustment not required in Adults, and if yes, why?  
> **username_1**: Age adjustment is not routinely done for adults. raw IQ is fairly stable over early-middle adulthood, whereas raw scores on IQ tests change a lot over childhood.  
> **username_3**: When you say "smoothing" does it mean something like running an OLS regression of the raw score on age (and maybe age-squared, age-cubed etc), take the residuals, and then center them to have mean 100 and sd 15?  
> **username_1**: @username_3; Yep. Something like that. Obviously, there's lots of options for the model both in what the mean is and whether and how you let the error variance vary over time. A major benefit is that you can get more granular predictions particularly over childhood where the difference between ages 7 and 0 months and 7 and 8 months would be meaningful.  

---

