Normal Distribution

Recap

• How to Summarize Data

• Descriptive Statistics

• Central Tendency:

  • Mean: The average value of a dataset.

  • Median: The middle value when data is organized in ascending order.

  • Mode: The most frequently occurring value(s) in a dataset.

• Dispersion:

  • Range: The difference between the highest and lowest values.

  • Variance: A measure of the spread of a set of values.

  • Standard Deviation (SD): A measure that quantifies the amount of variation or dispersion of a set of values.

• What does SD mean?: It tells us how much individual scores in a dataset tend to deviate from the mean.

Histogram

• Visual tool to represent frequency distribution of data.

  • Illustration of Weights Histogram: (shows frequency distribution with weights on the x-axis ranging from 132.5 to 148.5)

  • Represents different count frequencies in defined ranges.

Frequency Curve

• Distribution: Overlay of a histogram with a normal distribution curve.

  • Visual representation may involve counts from various data bins.

Types of Distributions

Unimodal vs. Bimodal

• Unimodal Distribution:

  • Contains one prominent peak in a histogram or frequency curve.

• Bimodal Distribution:

  • Contains two prominent peaks.

Skewed Distributions

• Definition: If one tail of the distribution is longer than the other.

  • Right-Skewed Distribution (Positively Skewed):

  • Characterized by a long right tail.

  • Higher values are more spread out, with many lower values.

  • The highest point is referred to as the mode.

  • The mean is located to the right of the median.

  • Left-Skewed Distribution (Negatively Skewed):

  • Characterized by a long left tail.

  • Lower values are more spread out, with many higher values.

  • The highest point is also the mode.

  • The mean is located to the left of the median.

Not Skewed: Symmetrical Distributions

• Defined as having no tail longer than the other.

• Key Characteristics:

  • Mean, mode, and median are equal.

  • The distribution is balanced around the mean.

Normal Distribution

• A special type of symmetrical distribution characterized as a bell-shaped frequency curve, often referred to as the Bell Curve.

• Commonly observed in naturally occurring phenomena including:

  • Example distributions: IQ, height, weight.

• Characteristics of Normal Distribution:

  • Majority of values are centered around the mean.

  • As values deviate from the center, their frequency diminishes.

Four Fundamental Characteristics of Normal Distribution

1. Symmetrical:

   • The upper and lower halves are mirror images.

2. Unimodal:

   • Mean, median, and mode coincide and are located at the distribution's peak.

3. Asymptotic:

   • The tails of the distribution approach but never touch the x-axis.

   • Indicates that the probability of extreme scores remains greater than zero.

4. Empirical Rules:

   • 68% of values fall within one standard deviation (${SD}$ ) from the mean.

   • 95% of values fall within two standard deviations ($2{SD}$ ) from the mean.

   • 99.7% of values fall within three standard deviations ($3{SD}$ ) from the mean.

   • Standard Deviation: A unit indicating the average distance of individual scores from the mean.

Function of Normal Distributions

• Many variables we examine are approximately normally distributed, which simplifies statistical analysis.

• Statistical tests are primarily built on the assumption of normality.

• When the mean and standard deviation of a normal distribution are known, it facilitates conversions between raw scores and percentiles.

Real-Life Applications

• Enables individuals to ascertain their relative standing within a distribution.

  • Example:

  • Exam score: 94

  • Class Mean: 80

  • Standard Deviation: 14

  • Interpretation:

    • “84% of students scored below you.”

    • “Only 16% of the students performed better.”

Raw Scores and Percentiles

• Example with IQ Scores:

  • Normal distribution: mean = mode = median, unimodal.

  • Here, M = 100, ${SD}=15$ .

  • Using the empirical rules:

  • “Half the population has an IQ of less than 100.”

  • “Half the population has an IQ of greater than 100.”

  • “68% of the population have IQs between 85 and 115.”

  • “95% of the population have IQs between 70 and 130.”

Relative Position within Output (Beyond Standard Deviations)

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• What about values that are not exactly one or two standard deviations from the mean?

  • Consider IQ values: 102, 99, 93.

  • Estimating one's relative standing requires calculations, leading to the necessity of standardized scores (discussed in the next lecture).

Summary

• Understanding distributions is critical, including:

  • Characteristics of unimodal, bimodal distributions.

  • Identifying left-skewed, right-skewed, and normal distributions.

  • Importance of understanding raw scores and their corresponding percentiles.