Descriptive statistics on the NCE: pick the right center, read the curve, convert the scores.
Three skills carry this topic. Choose the right center. Read the normal curve. Convert between standard scores. Learn the arithmetic once and it works on every item that needs it.
By Dr. Andrew Wichterman · October 20, 2026 · 6 min read
AI-narrated overview made with Google NotebookLM. The written article below is by Dr. Andrew Wichterman.
Statistics is the part of the exam many counselors skim. That is a mistake, because these items reward a small set of facts and a little arithmetic. They ask you to choose a measure of center for a given distribution, read the normal curve, and convert one standard score into another.
Here is the working version of each skill, in the order a question tends to force you to use them. For the research side of statistics, including a short look at the normal curve, see Research and statistics on the NCE. This post goes deeper on the numbers.
Read it with a pencil. The conversions only stick once you have written them out yourself.
The practice item in this article was written for this site in the style of the NCE. It is not a question from the actual exam. NBCC does not release its items, and candidates sign a nondisclosure agreement.
Start with the scale of measurement
Before you pick any statistic, ask what kind of number you are holding. The scale decides which statistics, and which tests later on, are legitimate. The mnemonic is NOIR.
- Nominal. Categories with no order, such as diagnosis or gender. The mode is the only center that makes sense.
- Ordinal. Rank order without equal gaps, such as class rank or a Likert item. Use the median.
- Interval. Equal gaps but no true zero, such as IQ or temperature in Fahrenheit. The mean works. Ratios do not.
- Ratio. Equal gaps and a true zero, such as weight, income or number of sessions. Every statistic applies.
Do not skip this step. A question that sounds like it is about the mean may really be about whether the mean is allowed.
Mean, median and mode, and the pull of the tail
The mean is the average, and extreme scores drag it. The median is the middle score. It is the better center when the distribution is skewed or has outliers, income being the classic case. The mode is the most frequent score. It is the only choice for nominal data, and a distribution can have two of them.
In a normal distribution the three centers land on the same spot. In a skewed distribution they separate, and the order depends on which way the tail points.
- Positive skew. The tail points right and most scores are low. Reading from the left, the order is mode, median, mean.
- Negative skew. The tail points left and most scores are high. Reading from the left, the order is mean, median, mode.
Exam tip. Locate the long tail before anything else. That gives you the skew, and the mean is always the center pulled toward it.
Variability: how spread out the scores are
The range is the highest score minus the lowest. It is unstable, because one odd score changes it. Variance is the average of the squared distances from the mean. The standard deviation is the square root of the variance, which puts spread back into the original units. That is why it is the measure of spread you will use most. A larger standard deviation means a more mixed set of scores.
The interquartile range covers the middle half of the scores, from the 25th to the 75th percentile. Outliers barely move it. Kurtosis is about the peak: leptokurtic is tall and thin, platykurtic is flat, and mesokurtic is the normal shape.
The normal curve and its fixed percentages
The normal curve is symmetric and bell-shaped, and the mean, median and mode all sit at the center. The percentages under it are fixed, so learn them exactly.
- About 68 percent of scores fall within one standard deviation of the mean, about 95 percent within two, and about 99.7 percent within three.
- Slice by slice: about 34 percent between the mean and one standard deviation, about 13.5 percent between one and two, and about 2 percent between two and three.
- Percentiles follow from that. One standard deviation above the mean is near the 84th percentile, two above is near the 98th, and one below is near the 16th.
Expect the percentages themselves to appear as answer choices. A near-miss number is a distractor you can eliminate once you know the real ones.
Standard scores: one idea, several scales
A z-score tells you how many standard deviations a raw score sits from the mean. Subtract the mean from the raw score, then divide by the standard deviation. A z-score has a mean of 0 and a standard deviation of 1. Every other standard score is the same idea on a different ruler.
- T-score. Mean 50, standard deviation 10, used with the MMPI. T equals 10 times z, plus 50.
- Deviation IQ. Mean 100, standard deviation 15.
- SAT and GRE subtests. Historically a mean of 500 and a standard deviation of 100.
- Stanines. Nine bands, with a mean of 5 and a standard deviation of about 2.
A percentile rank is the percent of scores at or below a given score. It is ordinal, not equal-interval, and percentiles crowd together in the middle of the distribution. That means a five-point gap near the 50th percentile reflects a smaller real difference than the same five points near the 95th.
Put the pieces together once and keep them: a z of plus 1 matches a T of 60, an IQ of 115, and about the 84th percentile.
How this shows up on the exam
Items in this area ask you to do one of three things: pick the right measure of center for a distribution, read the normal curve, or convert among standard scores. Work them in a fixed order.
- Name the scale of measurement.
- Check for a tail, and let it tell you the skew and where the mean sits.
- If scores come from different tests, convert them to z before you compare them.
The tempting wrong answers follow a pattern. The mean tempts when the data are skewed. A percentile tempts when the question is really about the size of a difference. A z of plus 1 tempts you toward the 68th percentile, because 68 percent is the share within one standard deviation on both sides. The share below plus 1 is about 84 percent. For the full lesson text, use the course lesson.
Try two
A clinic reports the mean and the median of its clients' household incomes. A few very high incomes are in the sample. Which statement best describes the distribution?
- Negative skew, with the mean below the median
- Positive skew, with the mean above the median
- Normal, with the mean equal to the median
- Positive skew, with the mean below the median
Show the answer
B. A few very high scores create a tail pointing right, which is positive skew. The mean is pulled toward the tail, so it sits above the median. Option D tempts because it names the right skew but reverses the order of the centers.
A client earns a deviation IQ of 115. Which pair describes the same score?
- z = 1, T = 60
- z = 1.5, T = 65
- z = 1, T = 50
- z = 0, T = 100
Show the answer
A. With a mean of 100 and a standard deviation of 15, an IQ of 115 sits one standard deviation above the mean, so z is plus 1. T equals 10 times z plus 50, which gives 60. Option C tempts because it keeps the z but treats the T mean of 50 as the score.
Check the current handbook. Exam logistics, fees and policies change. Confirm specifics against the NBCC candidate handbook and your state board's current rules. Aligned to the current NCE blueprint, valid for exams through June 30, 2027.
Name the scale. Find the tail. Convert with z. Then answer.