I know how to combine two observed values with different standard deviations into one mean value:

$$ \left(\frac{\sigma_b^2}{\sigma_a^2+\sigma_b^2}\right)x_a + \left(\frac{\sigma_a^2}{\sigma_b^2 + \sigma_a^2}\right)x_b = \mu $$

With a new standard deviation of:

$$ \frac{1}{\sigma_{new}} = \frac{1}{\sigma_a} + \frac{1}{\sigma_b} $$

How do I combine three values, $(x_a, x_b, x_c)$ with three different standard deviations, $(\sigma_a, \sigma_b, \sigma_c)$ into one mean, $\mu$? And what is that value's standard deviation?

Edit, here is some context:
I am reading the first chapter of Maybeck (1979) Stochastic models, estimation, and control (pdf). The equations above are 1-3 and 1-4 from page 12. My situation is that there is a red ball 5 ft in front of a robot that has three distance sensors: an IR laser, a visible spectrum laser, and a sonar. Each sensor, no matter how well crafted, will deliver a value estimate of the ball's distance from the sensor that will vary around 5 ft, over time, with a Gaussian distribution. So one sensor might read 4.9 ft, and vary with a std of 0.1 foot, over time. Another is poorly calibrated and reads 5.2 feet with a std of .5 feet, and the last reads 5.14 with a standard of 0.3 feet. (Those are just made up numbers given to illustrate the idea.)

More generically, lets assume there are three distances: x1, x2, and x3, with three standard deviations: sigma1, sigma2, and sigma3. Then you should be able to combine the distances x1, x2, and x3, using the SDs, mathematically to calculate a new mean which will essentially be 5.00001 feet, +/- 0.00002 std. I need the formula that lets me combine all three.

  • $\begingroup$ So, you want the mean of your data to be an unbiased estimate of the mean of the three populations, where the populations are of the same size (relative to each other) & their variances are known a-priori, is that right? $\endgroup$ – gung Dec 20 '15 at 20:40
  • $\begingroup$ Ok, that is close. Each of the three estimates, xa,xb,xc is an estimate of the size of a single population. Each estimate has an error of $\sigma_a,\sigma_b,\sigma_c$... I need the final mean and standard deviation given all three observations... (the final standard deviation should be smaller than all three initial standard deviations) $\endgroup$ – donlan Dec 20 '15 at 20:53
  • $\begingroup$ Do you mean "the size..." as in a count (N)? Eg, the number of fish in this pond? $\endgroup$ – gung Dec 20 '15 at 20:56
  • $\begingroup$ So, the example I am working from is a navigation analogy. Say you are on a ship, and three people take three readings on the ship's location using the stars. Each person is differently skilled, they each get different results, and each has a different standard deviation. What is the adjusted mean--given each person's measurement and standard deviation--position (imagine your position is 1D, for simplicity) $\endgroup$ – donlan Dec 20 '15 at 20:59
  • $\begingroup$ in our example, probably the number of individuals in the population. But, anyhow, it is generic... it is just three conclusions drawn about something 1-dimensional and uncertain, with a mean and standard deviation corresponding to each conclusion. $\endgroup$ – donlan Dec 20 '15 at 21:04

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