Final answer:
Normal or bell-shaped distributions are tall in the middle and short on the sides, and they are symmetric.
Step-by-step explanation:
Normal distributions, also known as bell-shaped distributions, exhibit a characteristic tall peak in the center and taper off gradually towards both ends. This shape signifies that most of the data points cluster around the mean, making it tall in the middle and shorter on the sides. The symmetry of the normal distribution implies that if you were to fold the graph along the vertical line at the mean, both sides would mirror each other perfectly.
Mathematically, the normal distribution is defined by its probability density function (pdf), often denoted as f(x). This function follows the formula:
f(x) = (1 / (σ√(2π))) * e^(-(x-μ)² / (2σ²))
Here, μ represents the mean and σ stands for the standard deviation. The peak of the bell-shaped curve occurs at the mean (μ) and the curve's standard deviation (σ) determines how spread out or narrow the curve is. In a normal distribution, approximately 68% of the data falls within one standard deviation from the mean, 95% within two standard deviations, and about 99.7% within three standard deviations due to its symmetric nature.
Normal distributions are extensively used in statistics due to their prevalence in nature and various phenomena, aiding in understanding and analyzing data across multiple fields, from social sciences to natural sciences and finance. Their distinctive characteristics of being symmetric and having a central peak make them a fundamental concept in probability and statistics.