1.5 Standard Deviations Below Mean (Stats Guide)
A value 1.5 standard deviations below the mean has a z-score of −1.5. Calculate it by subtracting 1.5 times the standard deviation from the mean. If the data follow an approximately normal distribution, this value is near the 6.68th percentile. Without that distribution shape, the z-score still shows distance from the mean, but not percentile rank.
A score can sound precise and still be easy to misread: “1.5 below average” might describe a distance, a rank, or a raw measurement, depending on the information available. The key is to separate those ideas. I’ll show how to find the value, check the calculation, and decide what you can safely conclude.
What does 1.5 standard deviations below the mean mean?
This phrase describes a value’s position relative to a mean. It means the value is one and a half standard deviations lower than the reference mean. The phrase alone does not tell you the raw value or its percentile; you need the mean, standard deviation, and, for a percentile estimate, a suitable distribution model.
A standard deviation measures how spread out values are around a mean. It uses the same units as the data. For example, if test scores are measured in points, the mean and standard deviation are also measured in points.
“Below” gives the direction. A value below the mean has a negative standardized score, called a z-score. So a value 1.5 standard deviations below the mean has a z-score of −1.5.
That does not mean the raw value is always 1.5. The number 1.5 tells you the distance in standard-deviation units, not the measurement itself. A score of 70 might be 1.5 standard deviations below one group’s mean, while a different group’s data could produce another raw score.
What formula finds the value?
To find the raw value 1.5 standard deviations below a mean, subtract 1.5 standard deviations from that mean. Use the population mean and standard deviation when those describe the full population; use the sample mean and sample standard deviation when working from a sample.
For a population, the formula is:
[ x=\mu-1.5\sigma ]
Here, (x) is the value you want, (\mu) is the population mean, and (\sigma) is the population standard deviation.
For a sample-based calculation, use:
[ x=\bar{x}-1.5s ]
In this formula, (\bar{x}) is the sample mean and (s) is the sample standard deviation. Keep the reference statistics together: do not combine a sample mean with a population standard deviation unless your course or analysis clearly explains why.
The related z-score formulas are:
[ z=\frac{x-\mu}{\sigma} \quad\text{or}\quad z=\frac{x-\bar{x}}{s} ]
Setting (z=-1.5) and rearranging gives the cutoff formulas above. The negative sign matters because the value is below the reference mean.
How do you calculate the raw value?
First identify the data and the right reference statistics. Then multiply the standard deviation by 1.5 and subtract that result from the mean. Keep the units, and check that the answer is lower than the mean.
Worked example using test scores
Suppose a class has a mean score of 80 points and a standard deviation of 6 points. The question asks for the score 1.5 standard deviations below the mean.
- Multiply the standard deviation by 1.5: (1.5 \times 6 = 9) points.
- Subtract 9 points from the mean: (80-9=71) points.
- Check the result: 71 is below 80, as it should be.
The answer is 71 points, or 1.5 standard deviations below the mean. If these are sample statistics, describe the result as 1.5 standard deviations below the sample mean, based on the sample standard deviation.
Check the result with the z-score formula
Substitute 71 into the z-score formula:
[ z=\frac{71-80}{6}=\frac{-9}{6}=-1.5 ]
The calculation confirms the answer. This reverse check is useful if you have several steps or are working with decimal values.
| Mean | Standard deviation | Calculation | Value 1.5 SD below |
|---|---|---|---|
| 80 points | 6 points | (80-(1.5\times6)) | 71 points |
| 100 units | 12 units | (100-(1.5\times12)) | 82 units |
| 50 seconds | 4 seconds | (50-(1.5\times4)) | 44 seconds |
These examples show why the standard deviation is essential. A mean alone cannot determine the cutoff. The same mean can lead to different values when the data have different spreads.
What percentile is a z-score of −1.5?
For a standard normal distribution, a z-score of −1.5 corresponds to a cumulative probability of about 0.0668, or the 6.68th percentile. This means about 6.68% of values fall at or below that point in the model. Treat this as a model-based estimate, not a rule for every dataset.
A percentile describes the share of values at or below a point. In a normal model, the curve is symmetric and follows a known shape. A z-score of −1.5 is therefore linked to an area of about 6.68% in the lower tail.
The central area between z-scores of −1.5 and +1.5 is about 86.64% in a standard normal distribution. This is the area between the two cutoffs, not the area below −1.5.
Use this distinction when reporting results:
| What you know | What you can say |
|---|---|
| The value has (z=-1.5) | It is 1.5 standard deviations below the reference mean. |
| The data are reasonably modeled by a normal distribution | It is near the 6.68th percentile in that model. |
| The data are not known to be approximately normal | Do not assign the 6.68th percentile based only on the z-score. |
| You have the full observed dataset | You can also find the observed rank directly, using the method your class or analysis specifies. |
For a continuous normal model, the probability of being strictly below the cutoff is the same as the cumulative probability at the cutoff. A single exact value has probability zero in that continuous model; the cumulative probability describes the area up to the cutoff.
When is the percentile estimate reliable?
A z-score of −1.5 always states a distance from the chosen mean in standard-deviation units. The percentile interpretation depends on the distribution’s shape. If the data are skewed, have strong outliers, or come from a small sample, the normal-model percentile may not match the observed rank well.
A normal distribution is a symmetric, bell-shaped model. When a dataset is reasonably close to this shape, the normal curve can give useful probability estimates. But a z-score by itself does not prove that the data are normal.
For instance, imagine income data with a few very high values. The distribution may be strongly skewed. A point 1.5 standard deviations below the mean still has a z-score of −1.5, but its actual percentile depends on the data’s shape. It may not be near the 6.68th percentile.
In a small sample, one or two values can also affect the mean and standard deviation. That can change both the calculated cutoff and how well a normal model represents the data. If you have the observations, compare the cutoff with their actual distribution rather than treating the normal estimate as certain.
How should you report the result?
State the calculation, reference statistics, and interpretation separately. That makes clear whether you are reporting a raw cutoff, a standardized distance, or a model-based percentile. Include units for raw values and qualify percentile estimates when they rely on a normal-distribution assumption.
A clear report might say: “Using a mean of 80 points and a standard deviation of 6 points, the cutoff 1.5 standard deviations below the mean is 71 points. Its z-score is −1.5. If the scores are approximately normally distributed, this cutoff is near the 6.68th percentile.”
If normality has not been checked, use a narrower statement: “The value is 1.5 standard deviations below the reference mean.” That is accurate without making an unsupported claim about rank.
Before submitting your answer, check these points:
- Did you identify whether the statistics describe a population or a sample?
- Did you use matching notation and statistics: (\mu,\sigma) or (\bar{x},s)?
- Did you subtract 1.5 times the standard deviation?
- Is the answer below the mean, with the right units?
- Did you give a percentile only when the normal model is appropriate?
Frequently asked questions
These short answers review the main calculations and interpretation limits. They distinguish the z-score from the raw value and explain when the normal percentile applies, so you can choose wording that fits the information you have.
What is the z-score for 1.5 standard deviations below the mean?
It is −1.5. A negative z-score means the value is below the reference mean.
How do I calculate the raw value?
Use (x=\mu-1.5\sigma) for population statistics or (x=\bar{x}-1.5s) for sample statistics.
What does a z-score of −1.5 mean?
It means the value is 1.5 standard deviations lower than the mean used to calculate the score.
Is −1.5 the same as the raw value?
No. The z-score is measured in standard-deviation units. The raw value depends on the mean and standard deviation.
What percentile is −1.5 in a normal distribution?
It is approximately the 6.68th percentile in a standard normal model.
Is it always the 6.68th percentile?
No. That estimate requires an approximately normal distribution. In other distributions, the actual rank can differ.
What percentage lies between z = −1.5 and z = +1.5?
About 86.64% lies between those points in a standard normal distribution.
Should I use the sample or population standard deviation?
Use the version that matches your reference data and course instructions. Pair (s) with a sample mean, or (\sigma) with a population mean.
Can the cutoff be negative?
Yes. If the mean is low enough relative to the standard deviation, subtracting 1.5 standard deviations can produce a negative raw value. Check whether that result makes sense for the variable.
How can I check my answer?
Put the raw value into the z-score formula. The result should be −1.5, allowing for rounding.
(This article was written by one of our staff writers, Alan Westbrook. Visit our Meet the Team page.)
