Compare observations from different distributions using z-scores

What is the first step when comparing observations from different distributions?

Compare two observations already standardised using their own group summaries. B has z=0.5. Move A from z=−2 to z=2; predict which is higher, lower, or closer to zero. Look especially at A=−0.5 and A=0.5

At A=−0.5 and B=0.5, what does the closest-to-mean rule say?

Comparing standard scores

Standard scores allow observations from different distributions to be compared on a common scale. Calculate each score first, then apply the direction that the context considers favourable

1. 1Standardise each observation Calculate a standard score separately for every distribution using its own mean and standard deviation \(\displaystyle z_i=\frac{x_i-\bar{x}_i}{s_i}\)
2. 2Compare standard scores Use standard scores rather than raw observations because the distributions can have different centres and spreads \(\displaystyle z_A=2,\quad z_B=1\quad\Rightarrow\quad z_A>z_B\)
3. 3Apply the favourable direction For a higher-is-better measure choose the larger z-score, and for a lower-is-better measure choose the smaller z-score \(\displaystyle \text{lower is better}\quad\Rightarrow\quad -2<-1\quad\text{is preferred}\)
4. 4Use distance when stated When closeness to the mean is preferred, compare the absolute standard scores and choose the smaller distance from zero \(\displaystyle |z_A|=1<|z_B|=2\quad\Rightarrow\quad A\text{ is closer}\)

Given values

\(\displaystyle x_A=78,\quad\bar{x}_A=70,\quad s_A=4\)

Group A has observed value 78, mean 70 and standard deviation 4. Find its standard score

Given values

\(\displaystyle x_B=78,\quad\bar{x}_B=72,\quad s_B=6\)

Group B has observed value 78, mean 72 and standard deviation 6. Find its standard score

Group A has observation 78, mean 70 and standard deviation 4. Group B has observation 78, mean 72 and standard deviation 6. A higher result is preferred. One of these comparisons is right. Which one?

State what a favourable result means

Standardise every observation with its own group mean and positive standard deviation. For a stated higher-relative-position criterion, rank z from largest to smallest. For lower-relative-position, rank from smallest to largest. For closeness to the group mean, rank |z| from smallest to largest. Keep ties

1. 1Limit the claim A standard score compares relative position in standard-deviation units. It does not automatically compare absolute achievement or imply equal percentiles in differently shaped distributions. State the criterion and the groups; changing the criterion can change the ranking

Given values

\(\displaystyle x=65,\quad\mu=68,\quad s=2\)

Group C has observation 65, mean 68 and standard deviation 2. Find its signed standard score

Two observations have standard scores −0.5 and −1.5. Which is closer to its own group mean?

One observation has x=82, mean 70, s=6. Another has x=90, mean 84, s=3. Under the higher-standard-score criterion, which is preferred?

Compare the same evidence under three criteria

Four fictional groups have the following observation, mean and standard deviation. Calculate all four signed standard scores. Rank them separately by higher relative position, lower relative position, and closest to the group mean. Preserve ties

1. 1Explain your claims Identify the largest raw observation and explain why it does not alone settle the relative ranking. State what further assumptions would be required to interpret z as a normal-model percentile. Do not describe lower numerical scores as universally better: this task explicitly compares three hypothetical criteria
2. 2Choose and verify technology Choose table and sorting functionality. Record one formula per row using that row’s mean and spread, the sort direction and whether the sorted quantity is z or |z|. Verify every result using the relationship below and check tied records are retained \(\displaystyle x=ar{x}+z imes s\)

Model response: the preference changes the order

The standard scores are A=2, B=2, C=−1.5 and D=−0.5. Higher-relative-position order is A=B, D, C. Lower-relative-position order is C, D, A=B. Closest-to-mean order is D, C, A=B.

1. 1Interpretation B has the largest raw observation, 90, but ties A after standardisation. A and B are each two standard deviations above their respective means. Their percentile ranks need not agree if their distributions have different shapes; the arithmetic does not establish normality or interchangeable group contexts
2. 2One reproducible workflow Use columns Group, Observation, Mean, SD, z and |z|. For each row calculate =(Observation-Mean)/SD and =ABS(z). Sort z descending, z ascending, then |z| ascending for the three criteria. Keep both rows in a tie. Check 70+2×6=82,84+2×3=90,68−1.5×2=65,60−0.5×6=57 independently

Two groups have equal z-values but different distribution shapes. What can you conclude?