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The five-point response list (e.g. "strongly approve"; "approve"; "undecided"; "disapprove", "strongly disapprove") for gathering data about attitudes - commonly but perhaps incorrectly called a Likert scale - was proposed in 1932 by Rensis Likert in "A Technique for the Measurement of Attitudes", Archives of Psychology, Volume 22, 5-55 (1932).

On page 22 of the article Likert describes one of the methods he used for scoring responses.

The percentage of individuals that checked a given position on a particular statement was converted into sigma values ... Table 22 of Thorndike's table greatly facilitated this calculation. These tables assume that one hundred percent of the cases fall between -3 and +3 sigma. The values given in the table are the average sigma values of intervals represented by the stated percentages, the origin considered to be at the mean.

("Thorndike" is a 1913 statistics textbook. Table 23 in Thorndike gives the kinds of values that would today be computed using, say, R's pnorm() function: e.g. 7.93% of a normally distributed population is between 0.0 and 0.2 standard deviations from the mean. Table 22 - the one Likert mentions - gives the same information, but in terms of average distribution values instead of standard distribution values. Even in 1913, sigma refered to the standard deviation, so it's confusing that Likert says he used Table 22 to compute his "sigma" values, when Table 23 would be more immediately relevant.)

Then Likert gives an example of his sigma-based scoring:

                         Strongly Approve Undecided Disapprove Strongly
                          Approve                             Disapprove
Percent checking            13%     43%     21%         13%       10%
Corresponding 1-5 value      1       2       3           4         5
Corresponding sigma value  -1.63   -0.43   +0.43       +0.99     +1.76

I think he assumes a normal distribution (skewed to the left in this example), and then assigns sigma values based on the cumulative distribution. But try as I might I can't reproduce Likert's sigma values.

Does anybody else want to give it a shot?

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1 Answer 1

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Thorndike's Table 22 displays the expected value of a doubly-truncated normal distribution, which can be seen as a conditional expectation given that the variate is in an interval specified by quantiles:

$$\mathbb{E}(Z \mid z_p<Z<z_{p+q}) = \frac{\phi(z_p)-\phi(z_{p+q})}{q}$$

where $z_p$ is the lower $p$th quantile of $Z\sim N(0,1)$, $\phi$ is the PDF of $Z$, and $0<p<1,\ 0<p+q<1$.

R-code for Likert's data:

E <- function (p, q) {(dnorm(qnorm(p)) - dnorm(qnorm(p+q))) / q}

P <- c(0.13, 0.43, 0.21, 0.13, 0.10)

p <- 0
for (q in P) {
  cat (p, q, E(p, q), "\n")
  p <- p + q
}

Output:

0     0.13  -1.62727 
0.13  0.43  -0.4252946 
0.56  0.21   0.4322558 
0.77  0.13   0.9857673 
0.9   0.1    1.754983

Online sources: Likert, Thorndike

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  • $\begingroup$ I now see that it would have helped if I had looked at Table 22 in the 2nd Edition of Thorndike, instead of the 1st Edition. $\endgroup$
    – Charles Eliot
    Commented Oct 1, 2016 at 6:18
  • $\begingroup$ @CharlesEliot - I made the same mistake at first; also, it took a while to understand how to read this table. Aside from all entries being integer percentages, it uses upper quantiles ${z^*_p}=z_{1-p}=\Phi^{-1}(1-p)$ instead of the more-common lower quantiles $z_p=\Phi^{-1}(p)$; that is, the table entry for (column $c$, row $r$) is actually $\mathbb{E}(Z\mid z^*_{p+q}<Z<z^*_p) = \mathbb{E}(Z\mid z_{1-(p+q)}<Z<z_{1-p}) = E(1-(p+q),q)$, where $p = \frac{c}{100},q=\frac{r}{100}$. E.g., $(c=5,r=11)\implies E(1-(0.05+0.11), 0.11) = 1.274328$. $\endgroup$
    – r.e.s.
    Commented Oct 1, 2016 at 14:20

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