Final answer:
In a normal distribution, data clustered near the mean suggests random, common causes of variation. The curve's shape on the mean and standard deviation. Large sample sizes reinforce the normality assumption as per the central limit theorem.
Step-by-step explanation:
If the measurements in a frequency distribution chart are grouped near the mean in a normal distribution, this implies that most of the variation is due to random, common causes rather than specific, unusual events or causes. In a normal distribution, data tends to cluster around the mean, and as the data moves away from the mean, it occurs less frequently, resulting in the familiar bell-shaped curve. The mean, median, and mode all lie at the same point in a normal distribution. Furthermore, the shape of the distribution is determined solely by the mean and the standard deviation.
The central limit theorem supports the notion that regardless of the original population distribution, the distribution of sample means tends to follow a normal distribution, particularly when large samples are considered. With the sampling distribution having the same mean as the original distribution and its variance being the original variance divided by the sample size. This normality assumption is critical in many statistical tests and inferences.
During a collaborative exercise where you record data and create histograms, a smooth bell-shaped curve should ideally emerge, providing that the sample sizes are sufficiently large (usually greater than 30 according to the central limit theorem). The means of each dataset would be placed on the x-axis under the peak of each curve, and areas under the curve would represent probabilities for particular measurements being above or below certain values if the total area is considered to be one.