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I am looking for help with statistical confidence levels. I have asked a court appointed company valuer a question, see below, in regard to factual information on multiples applied to a company's EBITDA (adjusted profits) to arrive at its value.

He claims there is an error but as the group that publish the multiples provide no raw data one cannot see how they arrived at this. I do not understand the confidence level of 95% that they give and how many samples you need to get it, in this case the price small companies sold for at a given date. Can the sample size be as low as five to give the range of 4.4 to 5.8 at 95% confidence or as the expert claims there would need to be a sample of at least 40?

I have read about confidence level and Confidence intervals but it goes above my head, I really need it in simple terms.

My Questions to the expert valuer In paragraph 8.76 of your report you stated that the UK200 Group survey of transactions is ‘perhaps the best evidence of actual deals taking place in the market at the smaller level.’ You did not state that at that time that you thought that there was ‘clearly an error’ in the statistics presented.

It is incorrect that there ‘is clearly an error’. The definition of median is the middle number in a group of numbers ranked in order of size. It is quite possible for the median (the middle number in a list of numbers sorted by size) also to be the lowest number. For example, say a group of EBITDA multiples were found to be:

3.6; 3.6; 3.6; 4.3; and 5.4. The median is 3.6 and the lowest multiple is also 3.6.

(M1) On what basis do you say that there is ‘clearly an error’?

Experts reply: The reason that there is clearly an error is because of the statistical data provided. The quote from the UK 200 Group article is as follows: “The standout statistic is the dramatic rise in EBITDA multiples, which are occurring (at a confidence level of 95%) in the range 4.4 to 5.8, a marked increase from the 2012 range of 3.6 to 5.4”. In order for the range to be given at a 95% confidence level there must have been at least two “outliers” (the highest and lowest values) that were excluded from the figures. Furthermore, for that to give a result that was accurate at a 95% confidence level would have to mean that there were approximately (or at least) 40 transactions in the sample, so that the range given for 38 out of those 40 transactions would be statistically accurate at a 95% (ie 38/40) confidence level. Put another way, the two “outliers” would be the 5% (ie 2/40) of transactions excluded from the sample. With at least 40 transactions in the sample, it is clearly an error that the median EBITDA multiple would be the same as the lowest EBITDA multiple, as it would require 19 out of those 38 transactions to have had an EBITDA multiple of exactly 3.6.

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    $\begingroup$ I've edited out the statement of urgency. I (we) don't doubt that your problem may be urgent to you, but that is an entirely private matter. There is no queue jumping or priority ordering here whereby stated urgency gets faster or better attention. If that's crucial to you, hire someone and pay them for making your question a high priority (not that CV includes any such hiring facility). $\endgroup$
    – Nick Cox
    Commented Oct 24, 2016 at 10:29
  • $\begingroup$ Seems like part of the question is, can there be a set of $x_i$ such that median is 4.5, the mean is $\bar{x} = \frac{1}{n} \sum_i x_i = 5.1$ and $4.4 = 5.1 + \frac{\sigma}{\sqrt{n}} F^{-1}(.025, n-1)$ where $F$ is the quantile function to the t-distribution with $n-1$ degrees of freedom. and $\sigma = \frac{1}{n-1} \sum_i( x_i - \bar{x})^2$. The question seems to reference this: uk200group.co.uk/news-and-blog/Blog/blog-41.aspx $\endgroup$
    – Matthew Gunn
    Commented Oct 24, 2016 at 11:21

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In order for the range to be given at a 95% confidence level there must have been at least two “outliers” (the highest and lowest values)

This quote shows a clear misconception about the 95%-confidence interval. What they are saying here and after this quote that the 95%-confidence interval of a series is defined by trimming the sorted list by 2.5% on both ends. This is just plain wrong.

The expert also poorly defined what the confidence interval actually is about:

the dramatic rise in EBITDA multiples, which are occurring (at a confidence level of 95%) in the range 4.4 to 5.8,

Is the confidence interval based on the median rise? That is not immediately clear from this quote, but lets presume it does.

The confidence interval then represents the bounds for the true median rise of EBITDA multiples, with a certainty of 95%. In other words, the data gives some sample median and based on that a confidence interval is constructed which has 95% probability of containing the true median.

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    $\begingroup$ In addition, the highest and lowest values are extremes, but they are not necessarily outliers in any statistical sense. What an outlier is precisely is contentious, but at the simplest level it would have to be a value much larger or smaller than others. Thus if there are three values 11, 12, 13 then 11 and 13 are the extremes but neither is an outlier unless exceptionally some independent, subject-matter expertise tells you that 11 or 13 would be very low or high) $\endgroup$
    – Nick Cox
    Commented Oct 24, 2016 at 11:54
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Consider the values:

\begin{bmatrix}4.5& 4.5& 4.5& 4.5& 5.625& 5.625& 6.45 \end{bmatrix}

The mean is 5.1. The median is 4.5. The sample standard deviation is .793. The number of observations is 7. The standard error is $.793 / \sqrt(7)$ = .3.

Let $F^{-1}(x, df)$ be the quantile function (i.e. inverse CDF) for the t distribution. $F^{-1}(.025, 6) = -2.45$. Hence the "95% confidence interval" is $[4.4, 5.8]$.

Purely computationaly, you can make it work out, but you'd want a larger sample than 7 to get the central limit theorem to kick in to make the use of the t-cdf an appropriate choice. (The median so far from the mean shows that there's significant skew in the underlying EBIDTA numbers and not normally distributed.)

Really, you'd want the raw data, the raw EBIDTA multiples to see if they did something reasonable.

Note: you can't get it to work out with simply 5 values.

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  • $\begingroup$ Number of samples is often scientist-speak whereas sample size is statistician-speak. $\endgroup$
    – Nick Cox
    Commented Oct 24, 2016 at 18:51
  • $\begingroup$ @NickCox Lol, yeah... It's also I wrote it at 6am and perhaps not quite awake speak. $\endgroup$
    – Matthew Gunn
    Commented Oct 24, 2016 at 19:04
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    $\begingroup$ As a non-statistician with statistical interests I often need to keep translating on this one to my colleagues and students who have samples of soil, rock, water, whatever. $\endgroup$
    – Nick Cox
    Commented Oct 24, 2016 at 19:25

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