T-Scores and Japanese Hensachi: How Deviation Scores Work

What a deviation score measures

A raw score of 80 means little on its own. On a test where the average is 50, it is excellent; on a test where the average is 75, it is only slightly above typical. Spread matters too. If almost everyone scored between 70 and 80, an 80 stands out more than it would on a test where scores ranged from 20 to 100.

A deviation score accounts for both the average and the spread. It is built so that an average result scores exactly 50, a result one standard deviation above average scores 60, and one two standard deviations above scores 70. Results below average fall under 50. Because every test is mapped onto the same scale, results from different tests become easier to compare.

In Japan this number is called hensachi and is a familiar part of school life: mock exam results, high school and university entrance difficulty, and prep school rankings are often expressed as hensachi. Statisticians elsewhere know the same transformation as a T-score.

The formula, step by step

The formula is: deviation score = (score - mean) / standard deviation x 10 + 50. The standard deviation measures how far scores typically sit from the mean. To get it, subtract the mean from each score, square the differences, average the squares (this is the variance), and take the square root.

Here is a small example. Five students score 50, 60, 70, 80 and 90. The total is 350, so the mean is 70. The differences from the mean are -20, -10, 0, 10 and 20; squared, they are 400, 100, 0, 100 and 400, which sum to 1,000. Dividing by 5 gives a variance of 200, and its square root, about 14.14, is the standard deviation.

The student who scored 80 has a deviation score of (80 - 70) / 14.14 x 10 + 50, about 57.1. The 90 becomes about 64.1, and the 50 about 35.9. On a larger test with a mean of 62 and a standard deviation of 12, a score of 80 gives (80 - 62) / 12 x 10 + 50 = 65, and a score of 50 gives 40.

• 1. Find the mean: total of all scores divided by the number of scores.
• 2. Square each score's difference from the mean and average them to get the variance.
• 3. Take the square root of the variance to get the standard deviation.
• 4. Compute (score - mean) / standard deviation x 10 + 50.

Reading a deviation score as a percentile

If scores follow a normal distribution, the symmetric bell curve, a deviation score maps to an approximate percentile. About 68% of a normal distribution lies within one standard deviation of the mean (scores of 40 to 60), and about 95% within two standard deviations (30 to 70).

Under that assumption, a score of 55 places you in roughly the top 31%, 60 in the top 16%, 65 in the top 6.7%, 70 in the top 2.3%, and 75 in the top 0.6%. A score of 40 means about 84% of the group scored higher.

• 55: roughly the top 31%
• 60: roughly the top 16%
• 65: roughly the top 6.7%
• 70: roughly the top 2.3%
• 75: roughly the top 0.6%

When the bell curve assumption breaks

Those percentiles only hold when the distribution really is close to normal. A test that is too easy bunches scores near the maximum; a test that is too hard bunches them near zero. In either case, the link between deviation score and rank weakens. Small groups have the same problem: with five or ten people, a score of 60 does not reliably mean the top 16%.

Deviation scores are often described as running from about 25 to 75, but there is no built-in limit. Take an extreme case where 99 out of 100 people score 0 and one person scores 100. The mean is 1 and the standard deviation is about 9.95, so the top scorer's deviation score is about 149.5. Heavily skewed data can push scores above 100 or below 0.

There is also a choice in how the standard deviation is computed. Dividing by the number of scores n gives the population standard deviation; dividing by n - 1 gives the sample standard deviation. Deviation scores for a whole class or test group normally use n. In the five-student example, dividing by n - 1 gives a standard deviation of about 15.81, and the 80 drops to about 56.3.

The group defines the score

A deviation score describes position within a particular group, so the same ability can produce different scores in different groups. Suppose two tests both have a mean of 60 and you score 70 on each. If the first test has a standard deviation of 10, your deviation score is 60. If the second has a standard deviation of 5, it is 70.

The same applies to the people taking the test. On an exam taken mostly by strong students, the mean rises, and a given raw score earns a lower deviation score than it would among a broader group. That is why Japanese mock exam providers' hensachi figures are not directly interchangeable. Before comparing deviation scores from different sources, check whether the test-takers and the test itself are comparable.

Trying it with the calculators

Paste a list of scores into the T-score calculator and it shows the mean, the standard deviation, and the T-score and rank of every entry. Enter an extra score to see where it would rank in that data set and what its T-score would be. The calculator uses the population standard deviation, dividing by n.

If you want to look more closely at averages themselves, the means calculator computes the arithmetic, geometric and harmonic means of a list of numbers. Deviation scores use the ordinary arithmetic mean, but other averages suit other jobs: the geometric mean fits growth rates, and the harmonic mean fits averages of rates such as speeds over equal distances.