Understanding the Normal Distribution

The question asks about the relationship between the Mean, Median, and Mode for a normal distribution. The normal distribution, also known as the Gaussian distribution, is a widely used probability distribution that is symmetric about its mean. Its shape is often described as a bell curve.

Properties of Normal Distribution

A key characteristic of a perfectly normal distribution is its symmetry. This symmetry has a direct impact on the measures of central tendency:

• Mean: The arithmetic average of all values in the dataset. It is the balance point of the distribution.
• Median: The middle value in a dataset when arranged in ascending or descending order. It divides the dataset into two equal halves.
• Mode: The value that appears most frequently in the dataset. It represents the peak of the distribution.

Relationship Between Mean, Median, and Mode in Normal Distribution

Due to the perfect symmetry of the normal distribution:

• The highest point of the curve (the peak) is where the Mode is located.
• Because the distribution is symmetric, the value that divides the area under the curve into two equal halves (the Median) is exactly at the center.
• The arithmetic average of the data points (the Mean) is also located precisely at the center of the distribution, pulling the values towards this central point.

Therefore, in a perfectly normal distribution, the Mean, Median, and Mode all coincide at the same central point.

Mathematically, this relationship is expressed as:

\( \text{Mean} = \text{Median} = \text{Mode} \)

Analyzing the Options

Let's look at the given options in the context of a normal distribution:

1. Mean < Median < Mode: This relationship typically occurs in negatively skewed (left-skewed) distributions, not symmetric normal distributions.
2. Mean > Median < Mode: This arrangement does not represent a standard type of distribution skewness.
3. Mean > Median > Mode: This relationship typically occurs in positively skewed (right-skewed) distributions, not symmetric normal distributions.
4. Mean = Median = Mode: This is the characteristic relationship for a symmetric distribution like the normal distribution.

Based on the properties of the normal distribution, the statement Mean = Median = Mode is true.

Conclusion on Normal Distribution Properties

For a theoretical, perfect normal distribution, the measures of central tendency are equal. In real-world data that approximates a normal distribution, these values will be very close to each other.

Revision Table: Normal Distribution Measures

Additional Information: Skewness and Kurtosis

While the normal distribution is characterized by symmetry, other distributions can be asymmetric (skewed) or have different peak/tail characteristics (kurtosis).

• Skewness: Measures the asymmetry of the probability distribution. A normal distribution has zero skewness.
• Kurtosis: Measures the "tailedness" and peakedness of the distribution. A normal distribution has a kurtosis of 3 (or 0 depending on the definition, excess kurtosis). Distributions with higher kurtosis have heavier tails and sharper peaks.

Understanding skewness helps explain why the Mean, Median, and Mode differ in non-normal distributions.