Question

One of the authors has an adopted grandson whose birth family members are very short. After examining him at his 2 -year checkup, the boy's pediatrician said that the \(z\) -score for his height relative to American 2-year-olds was \(-1.88\). Write a sentence explaining what that means.

Short answer

The boy's height is 1.88 standard deviations below the average height for American 2-year-olds, indicating he is shorter than most of his peers.

Step-by-step solution

Understanding Z-Score

The z-score is a statistical measurement that describes a value's position relative to the mean of a group of values. It is expressed in terms of the number of standard deviations the value is from the mean. A z-score of 0 indicates that the data point's score is identical to the mean score.

Interpreting Negative Z-Score

In this context, the z-score is negative, which means the boy's height is below the average (mean) height for American 2-year-olds. Specifically, a z-score of -1.88 means his height is 1.88 standard deviations below the mean height.

Understanding Standard Deviations

Being 1.88 standard deviations below the mean suggests that the boy is shorter than most of his peers. For standard normal distributions, about 95% of data lies within two standard deviations from the mean, which means this child's height is lower than approximately 97% (as 1.88 is close to 2) of 2-year-old American children.

Key concepts

Standard Deviations

Standard deviation is a measure of how spread out numbers are in a data set. It tells us the average distance of each data point from the mean. In simpler terms, it helps us understand how variable or consistent the data is.
When it comes to heights, a small standard deviation means most children are close to the average height. A large standard deviation indicates more variety in children's heights.

• It helps compare individual data points with the group.
• The further a child's height deviates from the average, the higher or lower the z-score will be.

Since a z-score indicates how many standard deviations a particular measurement is away from the mean, it provides quick insight. In this example, the boy's z-score of -1.88 tells us he's 1.88 standard deviations below average. This measurement is essential in understanding how his height compares with other children his age.

Mean Height

The mean height is the average height of a group of children. To calculate this average, you add up all the individual heights and then divide by the number of children. This mean gives us a central value around which other individual heights are compared.
Understanding mean height is crucial when interpreting z-scores, as a z-score is always in relation to this mean.

• If a child's height is equal to the mean, their z-score is zero.
• Heights taller than the mean have positive z-scores.
• Heights shorter than the mean have negative z-scores.

In the case of the boy with a z-score of -1.88, the mean height represents the typical height for his age group. His height is notably lower than this average.

American 2-Year-Olds

American 2-year-olds are typically studied to understand growth patterns and establish standards in pediatrics. Averages and standard deviations from a large sample of 2-year-olds provide benchmarks for comparing individual growth.
By using a z-score, doctors and parents can easily see how a child's growth compares to peers. Since the z-score in the example is -1.88, it tells us the boy is shorter than most American 2-year-olds.

• This figure suggests he is as tall or taller than only about 3% of the population his age.
• The remaining 97% are taller.

These comparisons allow for early assessments of growth issues or reassure average growth trends. Using American 2-year-olds as a reference helps ensure assessments are specific and relevant to the demographic in question.