Describing Graphed Distributions

Stem-and-Leaf Plots: A device for displaying quantitative data where each data value is split into a "stem" (the leading digit) and a "leaf" (the trailing digit).
Histograms: A graphical representation showing the frequency distribution of a dataset by using bars.
Polygons: A line graph representing the frequency of data points, connected at the midpoints of the histogram's bars.

Central Tendency: A measure that represents the center or typical value of a dataset. Common measures include mean, median, and mode.
Dispersion (Variability): The extent to which data points in a dataset differ from each other and from the mean. Common measures include range, variance, and standard deviation.
Skewness: A measure of the asymmetry of the probability distribution.
- Positive skewness indicates that the tail on the right side of the distribution is longer or fatter than the left.
- Negative skewness indicates that the tail on the left side is longer or fatter than the right.
Kurtosis: Refers to the "peakedness" of a distribution, characterizing the shape of the distribution's tails.
- Lepotokurtic: Distribution is thin and peaked ($ ext{Pearson kurtosis} > 3$).
- Mesokurtic: Distribution is moderate in shape ($ ext{Pearson kurtosis} = 3$).
- Platykurtic: Distribution is flat ($ ext{Pearson kurtosis} < 3$).
Modality: Refers to the number of peaks (modes) in the distribution. A distribution can be unimodal, bimodal, or multimodal.
Outliers: Data points that differ significantly from other observations, potentially influencing the mean and other statistical measures.

Scores: Scaled from 0 to 150.
Example Scores: 38, 53, 54, 58, 60, 62, 64, 65, 68, 70, 71, 73, 74, 76, 77, … , 147

Structure: Displays the frequency ( ext{number of occurrences}) of specific scores alongside their cumulative percentages.
Columns in Table:
- Valid Frequency: The count of data points for given scores.
- Percent: Percentage representation of the valid frequency in relation to total observations.
- Cumulative Percent: Running total of the percentages, showing the percentage of scores that fall below a particular value.

Score	Valid Frequency	Percent	Cumulative Percent
38	1	0.6	0.6
53	1	0.6	1.1
54	1	0.6	1.7
…	…	…	…
147	1	…	…

Histograms: Graphical representation that shows the distribution of scores by dividing the data into bins and counting the frequency of scores in each bin.
Polygons: Similar to histograms but display the frequency using a connected line across the midpoints of each bin.

Characteristics: Most graphed distributions are discussed in relation to the normal distribution, characterized by its bell-shaped curve.
Natural Variables: Many naturally occurring variables, such as height, weight, and IQ, tend to follow a normal distribution.
- Example: Individuals with extremely high or low scores are rare, whereas most people cluster around the average.

When describing a graphed distribution, refer to the following:

Kurtosis: Assess the peakedness (or flatness) of the graph.
Dispersion: Consider the spread or variability of the data.
Presence of Outliers: Evaluate if any extreme values deviate significantly from the rest.
Skewness: Consider if the distribution leans to one side (positive or negative skew).
Modality: Identify the number of peaks in the distribution.

Kurtosis Definition: Refers to how tall and sharp the peaks of a distribution are compared to a normal distribution.
- Lepotokurtic Distribution: Tall and thin; $ ext{Pearson kurtosis} > 3$.
- Mesokurtic Distribution: Moderately peaked; $ ext{Pearson kurtosis} = 3$.
- Platykurtic Distribution: Flat; $ ext{Pearson kurtosis} < 3$.

Definition of Skewness: The degree of asymmetry of a distribution.
- A positively skewed distribution: The tail on the right is longer; $ ext{mean} > ext{median} > ext{mode}$.
- A negatively skewed distribution: The tail on the left is longer; $ ext{mean} < ext{median} < ext{mode}$.

To effectively describe the distributions observed in data:

Assess the central tendency, variability, presence of outliers, kurtosis, skewness, and modality.