Explain the concept of normal distribution. Explain divergence from normality

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Normal Distribution:
The normal distribution, also known as the Gaussian distribution or bell curve, is a symmetrical probability distribution that is characterized by a bell-shaped curve.

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In a normal distribution:

  1. The mean (average), median, and mode are all equal and located at the center of the distribution.
  2. The distribution is symmetric, meaning the left and right sides of the mean are mirror images of each other.
  3. About 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

The normal distribution is a theoretical concept used in statistics, and many natural phenomena tend to follow this distribution, such as the distribution of heights, weights, IQ scores, and measurement errors.

Divergence from Normality:
Divergence from normality refers to situations where the data does not conform to the characteristics of a normal distribution. There are several ways in which data may diverge from normality:

  1. Skewness: A normal distribution is symmetric, but if the data is skewed, it means that the distribution is not symmetrical. Positive skewness indicates a longer right tail, while negative skewness indicates a longer left tail.
  2. Kurtosis: Kurtosis measures the “tailedness” of a distribution. A normal distribution has a kurtosis of 3 (mesokurtic). Higher kurtosis (leptokurtic) indicates heavier tails, and lower kurtosis (platykurtic) indicates lighter tails.
  3. Outliers: Extreme values, or outliers, can significantly affect the normality of a distribution. Outliers may cause skewness or kurtosis, impacting the symmetry and tail behavior.
  4. Multimodality: A normal distribution is unimodal (has one peak). If a distribution has more than one peak, it is multimodal and diverges from normality.
  5. Heavy Tails: Tails that are heavier than those of a normal distribution can affect the overall shape. This is often observed in distributions with extreme values or in fat-tailed distributions.
  6. Non-Linearity: In some cases, the relationship between variables may be non-linear, leading to a departure from normality.

When data diverges from normality, it may impact the validity of statistical analyses that assume normal distribution, such as parametric tests like t-tests or analysis of variance (ANOVA). In such cases, non-parametric tests or data transformation techniques may be considered. It’s important for researchers to assess the normality of their data before applying statistical methods that assume a normal distribution and, if necessary, explore alternative approaches that are robust to deviations from normality.