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The mean, often referred to as the average, is a crucial concept in statistics that represents the central point of a data set. To calculate the mean, you sum all the measurements and then divide by the number of observations. For instance, if we're considering the birth lengths of children, we add up all the individual lengths and then divide by the total number of children measured.

In the provided exercise, the mean birth length for U.S. children born at full term is presented as 52.2 cm. This number is the result of adding together all the birth lengths recorded and dividing by the count of those measurements. It's a measure that can give us a general idea about the size of newborns and serves as a benchmark to compare individual birth lengths.

Statistical range is a term that defines the difference between the smallest and largest values in a data set. However, in the context of our exercise, when we talk about range around the mean, we're particularly interested in the range of values within one standard deviation from the mean. Standard deviation is a statistic that measures the dispersion of a dataset relative to its mean and is calculated as the square root of the variance.

In simpler terms, standard deviation tells us how spread out the data points are. A small standard deviation indicates that the data points tend to be very close to the mean, while a high standard deviation indicates that the data points are spread out over a large range of values. When we calculate the range of birth lengths within one standard deviation from the mean (52.2 cm ± 2.5 cm), we get 49.7 cm to 54.7 cm; this range encompasses the middle portion of the distribution and is where a significant number of lengths are expected to fall.

Unimodal distribution refers to a frequency distribution that has a single clear peak or mode. This means that there's one value that occurs most frequently, which, in the case of a symmetrical and unimodal distribution, typically aligns with the mean and median of the data.

A unimodal distribution indicates the presence of a typical or common state, such as the most common birth length for full-term U.S.-born children. The symmetry in unimodal distributions shows that the data points are evenly distributed around the mean. When plotting the data, its graph takes the shape of a bell curve, which is known as a normal distribution. This pattern suggests that most newborns’ lengths will be near the average length of 52.2 cm, with fewer infants being on the extreme ends of the spectrum. This shape is useful in making predictions about the data and understanding the likelihood of different outcomes.