To determine which data set has the largest standard deviation, we need to understand what standard deviation measures and consider the characteristics of each data set provided.
Standard deviation is a statistic that measures the dispersion or spread of a set of values. A larger standard deviation indicates that the values are more spread out from the mean (average), while a smaller standard deviation indicates that the values are closer to the mean.
Now let's analyze each data set:
1. The collection of heights, measured in inches, of 1st- through 6th-grade students in an elementary school:
Students in grades 1 through 6 range in age from approximately 6 to 12 years old. During this time, children are growing at different rates, and there is a significant variability in height between the ages due to growth spurts and differences in the onset of puberty. Therefore, we would expect a larger variance in heights and consequently a larger standard deviation.
2. The collection of heights, measured in inches, of all 12th-grade students in a high school:
By the time students reach the 12th grade, most of them have completed their major growth spurts, and the heights of students within this age group are expected to be more consistent. While there will still be variability in heights (tall, average, and short students), this variation is generally not as extreme as in the case of younger students of varying ages. Therefore, the standard deviation of heights for 12th-grade students would likely be smaller than that of grades 1 through 6.
3. The length in inches of newborns at a hospital in one year:
Newborn lengths are relatively consistent because genetic factors play a large role in the length of a baby at birth, and environmental factors have not yet had significant influence. With less variability, the standard deviation for newborn lengths would be smaller than that for a diverse age group like 1st- through 6th-grade students.
Considering all these factors, the data set with the largest expected standard deviation would be the heights of 1st- through 6th-grade students, as this represents the group with the greatest age range and, consequently, the greatest variability in growth and development.
