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a and b are easy like you said, just display the value of the single measurement.
c now, how about a message like "Not applicable, only one measurement recorded." You don't show blank and you explain why they aren't getting a value.
I’ve given up trying to be calm. However, I am open to feeling slightly less agitated.
I’m begging you for the benefit of everyone, don’t be STUPID.
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Value has no meaning without time.
1) Never had a value.
2) last value at xx:xx:xx
3) etc.
Transparency and visibility lets me sleep at night.
(Also ... an early warning system for the attendant).
"Before entering on an understanding, I have meditated for a long time, and have foreseen what might happen. It is not genius which reveals to me suddenly, secretly, what I have to say or to do in a circumstance unexpected by other people; it is reflection, it is meditation." - Napoleon I
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Perhaps it will be helpful to the users to see displayed the number of said measurements.
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I see two alternative approaches
- Always show current measurement value, but show statistical values only when statistics is meaningful (you have a 'fair' number of samples).
or- Always show all values, however, at first measure: show the current value, omit or show the mean value (equal to current value) and omit the standard deviation.
my two cents.
"In testa che avete, Signor di Ceprano?"
-- Rigoletto
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Mathematicians define the (so called "Arithmetic") Mean as "Sum of all values divided by Count of all values" - see e.g. Arithmetic mean - Wikipedia[^]. Thus, the mathematically defined (Arithmetic) Mean for a single value is that single value. It does not at all matter whether you think whether this "makes sense" 
However, did whoever gave you this specification ("need to show ... Mean Value ...") specify that he/she actually
- meant you to use the arithmetic mean? (most probably yes; but asking back is not wrong - might only confuse the analyst who maybe doesn't even know that there are many other "Means");
- wanted to help the user also in such boundary cases? Maybe the analyst would even say "Oh, but for less than a handful of values that mean doesn't make any sense - there will always be 100 or 1000s of values, in practice"; then, you two should talk for some time what's the actual reason for showing those values ... That's what I think your "feeling" is about - you question the business value of showing the Mean (and Deviation) in boundary cases - which does matter.
H.M.
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If there is only one measurement, I would show something to indicate that measurement has begun with the opening value given and don't show any values for a, b, or c.
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Know your audience. Ask the user who will be looking at these values what they think they should see. The answers might surprise you.
For example, I have an application where I don't display statistical data until a certain number of 'stable' samples have been seen. Stability is determined by a filter on the input data, which isn't useful while the process being measured is starting.
Software Zen: delete this;
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the question for more context is that it depends.
so go back to the user and ask for more clerification what is useful to them
or what is the end goal of said mesaurement.
a number on it's own is meaningless
Someone needs to know if value is within tolerance - so current value unless said tolerance is based on a time window and so would need to be able to clearly indicate what that time window is used over the mean
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How often do values update? Once every 0.1 second? Once per second? Minute? Hour? Day? In the first two cases, display probably doesn't matter. If user has more than a few seconds of looking at the display, then it matters. (And how big is the display? "blank" you'd have space for, but I'm guessing not a "can't display value due to insufficient data" message would be too large.
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I would really like to add to your confusion. You're talking about sample count = 1 etc, but you're talking of a continuous stream of samples, have you thought about what "average" means if you have had 94760434786353845 samples accumulated over 72 hours in a changing signal, would you *seriously* want to report the average of all that?
My engineering self wants to suggest that you use a sliding average window of a fixed size, say, 10, but it could be anything reasonable considering the rate of change, and any noise that you would otherwise want to average out.
Then, if you pick that number, say, N=10, then you can only start reporting after you've had at least 10 samples. And that would answer your original question.
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I'd say it is an exception case that yields "Insufficient Data". One measurement is a data point. You can't infer any unbiased statistics when n=1. There is no average, mean, or SD.
If you're stuck having to show a number for some reason, that's an issue. I suppose you could treat it as a null, the same as having no data.
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For zero. Don't show anything. So don't even show the text.
Either for a report or a dashboard instead of those values display "No measurements"
For one, the calculated values still work. Presumably you also display a count of values.
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...One person can change the world[^] 
"I have no idea what I did, but I'm taking full credit for it." - ThisOldTony
"Common sense is so rare these days, it should be classified as a super power" - Random T-shirt
AntiTwitter: @DalekDave is now a follower!
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I prefer this one[^]
M.D.V. 
If something has a solution... Why do we have to worry about?. If it has no solution... For what reason do we have to worry about?
Help me to understand what I'm saying, and I'll explain it better to you
Rating helpful answers is nice, but saying thanks can be even nicer.
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"Dripping water hollows out stone, not through force, but through persistence." - Ovid, Roman poet
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Quote: Manjhi started off his extraordinary task in 1960, after his wife was injured while trekking up the side of one of the rocky footpaths. To reach the nearest hospital, he had to travel around the mountains, some 70 kilometers.
This is the very definition of what we do. Someone has an issue and we try to create a solution to fix it. So long as there are not too many stakeholders, committees, accountants, etc...
Graeme
"I fear not the man who has practiced ten thousand kicks one time, but I fear the man that has practiced one kick ten thousand times!" - Bruce Lee
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Now I know where Stephen King got the idea for Rita Hayworth and the Shawshank Redemption.
I’ve given up trying to be calm. However, I am open to feeling slightly less agitated.
I’m begging you for the benefit of everyone, don’t be STUPID.
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... you're a really crappy Secret Santa.
"I have no idea what I did, but I'm taking full credit for it." - ThisOldTony
"Common sense is so rare these days, it should be classified as a super power" - Random T-shirt
AntiTwitter: @DalekDave is now a follower!
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You spelt carpy rong. 
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If you give me a fresh 5 kg wild salmon, I'll be happy.
Religious freedom is the freedom to say that two plus two make five.
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Your cat is sending messages again.
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In America it's spelled Crappie[^]
I’ve given up trying to be calm. However, I am open to feeling slightly less agitated.
I’m begging you for the benefit of everyone, don’t be STUPID.
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