Calculate a standard score or actual value

Which formula calculates the standard score of an observation?

The mean is 50 seconds and standard deviation is 4 seconds. Predict the observation when z is negative, zero or positive, then move z from −3 to 3. Compare x with the fixed mean

What happens as z moves from −3 through zero to +3?

Calculating standard scores

A standard score measures how far an observation is from its mean in standard-deviation units. The sign shows whether the observation is below or above the mean

1. 1Find the signed deviation Subtract the mean from the observed value, keeping the order observed value minus mean \(\displaystyle x-\bar{x}\)
2. 2Divide by the standard deviation Use the standard deviation as the denominator to measure the deviation in standard-deviation units \(\displaystyle z=\frac{x-\bar{x}}{s}\)
3. 3Keep the sign A negative standard score is below the mean and a positive standard score is above the mean \(\displaystyle x<\bar{x}\Rightarrow z<0,\quad x>\bar{x}\Rightarrow z>0\)
4. 4Reverse the formula when needed When the standard score is known, multiply it by the standard deviation and add the signed change to the mean \(\displaystyle x=\bar{x}+zs\)

Given values

\(\displaystyle x=74,\quad\bar{x}=62\)

An observation is 74 marks and the mean is 62 marks. Find the signed deviation from the mean

Given values

\(\displaystyle x-\bar{x}=12,\quad s=6\)

The signed deviation is 12 marks and the standard deviation is 6 marks. Find the standard score

An observation is 74 marks, its mean is 62 marks and standard deviation 6 marks. Which numerator order gives its standard score?

Use a common unit and a positive spread

Use a strictly positive standard deviation. A standard score has no physical unit because the numerator and denominator use the same unit. A score of zero is at the mean; negative scores are valid. Calculating a standard score does not require normality, but normal-model percentages need an additional distributional assumption

1. 1Reverse and check Retain the sign when recovering the observation. Substitute the recovered value back into the forward formula. If all observations are equal, the standard deviation is zero and the ordinary standard score is undefined; do not divide by zero \(\displaystyle x=\bar{x}+zs\)
2. 2Precision Keep stored values until the final requested rounding. Parenthesise the whole numerator in a calculator or spreadsheet. A rounded z is an approximation and may not reconstruct the original value exactly

Given values

\(\displaystyle x=44,\quad\mu=50,\quad s=4\)

An observation is 44 seconds, mean 50 seconds and standard deviation 4 seconds. Find z.

Given values

\(\displaystyle \mu=50,\quad s=4,\quad z=2.25\)

For the same mean 50 seconds and standard deviation 4 seconds, recover the observation with z=2.25

Given values

\(\displaystyle x=48,\quad\mu=50,\quad s=3\)

An observation is 48 seconds, mean 50 seconds and standard deviation 3 seconds. Give z to two decimal places

Every observed value is 12, so the sample standard deviation is zero. What is the ordinary standard score of 12?

Calculate, reverse and verify standard scores

A fictional timing dataset has stated mean 50 seconds and standard deviation 4 seconds. Calculate standard scores for 44,50,59 seconds, interpret each sign and recover values for z=−2 and z=2.25. Do not assume a normal distribution

1. 1Units and precision Convert the mean, standard deviation and all three observations consistently into minutes and show that each z is unchanged. Separately use x=48, mean=50 and standard deviation=3 seconds to compare exact z with its two-decimal-place report and the value reconstructed from that report
2. 2Choose and verify technology Choose a calculator or spreadsheet and record the complete parenthesised formula. Verify every forward result by reversing it, and explain why z alone gives no empirical-rule percentage without a suitable model

Model response: preserve the sign and scale

Calculate each standard score from the observation, mean and standard deviation. The results place the observations below the mean, at the mean and above the mean. Reverse each result to check it, then recover the two requested values

1. 1Unit check Divide every time by 60 to convert seconds to minutes. The common factor cancels, giving the same three standard scores. Converting only the observation would mix units and invalidate the result \(\displaystyle \bar{x}=\frac{5}{6},\quad s=\frac{1}{15},\quad x=\frac{11}{15},\frac{5}{6},\frac{59}{60}\text{ minutes}\)
2. 2Precision check For the separate calculation, reversing the exact standard score gives 48 seconds; reversing its two-decimal report gives 47.99 seconds. Keep the exact or stored result for further work \(\displaystyle z=-\frac{2}{3}\approx-0.67,\quad50+(-\frac{2}{3})(3)=48,\quad50-0.67(3)=47.99\)
3. 3One reproducible workflow On a calculator, enter (44-50)/4 to obtain −1.5, then enter 50+(-1.5)*4 to check that the result is 44 seconds. No normality assumption was supplied, so these standard scores alone do not establish percentiles or 68–95–99.7% proportions

A negative standard score is obtained for a positively measured time. Is that a contradiction?