# Tag Info

1 vote

### Estimating mean and SD given the median and IQR values

The long answer is "no" this is not possible to determine without additional information about the distribution of the data. The pragmatic answer is "yes", if you make assumptions ...
• 5,564
1 vote
Accepted

### Non-parametric one-sample mean test for a bounded variable (based on Chebyshev's inequality?)

I think for this there are several well known concentration inequalities that can be applied. In particular Hoeffding, Azuma, and McDiarmid. Not that there's any real difference between the bounds one ...
• 1,624

### Why does a 95% Confidence Interval (CI) not imply a 95% chance of containing the mean?

The misinterpretation of confidence intervals is related to what Blitzstein and Hwang (in their probability textbook) call "sympathetic magic". Sympathetic magic is an anthropology term for ...

### Non-parametric one-sample mean test for a bounded variable (based on Chebyshev's inequality?)

Here is a maximum likelihood approach. Suppose you have $n$ observations of $X$ which total $T$, and you want to test the null hypothesis $E[X]=\mu$, with $T/n<\mu$. Then the maximum-likelihood ...
• 5,302
1 vote
Accepted

### Why can the standard error of the weighted mean be smaller than the standard errors of the individual measurements?

The short answer, as @LulY pointed out in the comments, is that you now have a larger combined sample size. Take the standard error of the mean for example: $$\mathrm{SE} = \frac{s}{\sqrt{n}}$$ Larger ...
• 12.7k
1 vote

### Why can't the slope be greater than 1 in regression to the mean for married couples' intelligence?

The slope of a linear regression is $$\text{slope} = \rho \cdot \frac{sd_y}{sd_x}$$ If we assume $sd_y \approx sd_x$ and $\rho<1$, then $\text{slope} < 1$. See also the question: Effect of ...
• 82.1k

### Why can't the slope be greater than 1 in regression to the mean for married couples' intelligence?

I interpret Kahneman as saying that you don't need to resort to theories about women marrying down if the correlation between husband and wife IQ is imperfect. But I suspect he was not precise in ...
• 37.2k
1 vote

### Why can't the slope be greater than 1 in regression to the mean for married couples' intelligence?

I think you are plotting the wrong thing. If you look at a fit of wife_iq vs. husband_iq, then we could well get a slope of 1, or less than 1, or more than 1. But what Kahnemann says is that highly ...
• 125k

### Under what conditions are there pairwise monotonic relationships between mean, variance, and (positive) skewness of a lower-bounded distribution?

It turns out there is a lower bound on the skewness of any strictly positive data set having given mean and sd: $$g_1 > \sigma/\mu - \mu/\sigma.$$ This doesn't seem entirely consistent with your ...
• 2,117

### Variance around true value, not mean

yes you can, if you do you may use the whole sample size as the denominator (n instead of n-1, as you aren't using any degree of freedom for estimating the mean), however, you may get a bad result if ...
• 4,555
Accepted

### Calculate mu and sigma of a log normal distribution from p50 and p99 in javascript

If $X \sim \operatorname{Normal}(\mu, \sigma^2)$ then $Y=e^X$ has a log-normal distribution with the same parameters and $X=\log(Y)$. So you simply need to take logarithms and exponentiation ...
• 40.5k
A log-normal variable is normally distributed in the log scale, so the identity holds in the log scale. To set the scene, let's review the normal case first. The $p^{th}$ quantile $Q(p)$ is calculated ...