Describing Distributions

Overview

When analysing data, it is important to describe the distribution, which provides a summary of how data values are spread. Key features of a distribution include shape, centre, and spread.

Key Features of a Distribution

Shape:

Symmetrical: Data is evenly distributed around the centre.
Skewed:
- Left-skewed: Long tail extends to the left; most data is on the right. In a left-skewed distribution, the mean of negatively skewed data will be less than the median.
- Right-skewed: Long tail extends to the right; most data is on the left. The mean of positively skewed data will be greater than the median, because positive outliers pull the mean higher.
Uniform: Data is evenly spread across the range.
Bimodal/Multimodal: Two or more peaks in the distribution.

Centre:

Measures of centre describe the typical value:
- Mean: The arithmetic average.
- Median: The middle value when data is ordered.
- Mode: The most frequently occurring value.

Spread:

Measures of spread describe variability in the data:
- Range: The difference between the maximum and minimum values.
- Interquartile Range (IQR): The range of the middle 50% of data.
- Standard Deviation: The average distance of data points from the mean.

Outliers:

Unusually high or low values that fall far from the rest of the data.

Visual Representation

Graphical tools help analyse the distribution:

Histograms: Show the frequency of data within intervals.
Box Plots: Display the spread, median, and potential outliers.
Stem-and-Leaf Plots: Provide a detailed view of individual values.

Examples

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Example 1: Identifying Shape

Problem:

A histogram of test scores shows most scores are concentrated at the higher end, with a few low scores. What is the shape of the distribution?

Solution:

The tail of the histogram extends to the left.
This indicates a left-skewed distribution. Answer: The distribution is left-skewed.

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Example 2: Calculating Spread

Problem:

Find the range and IQR of the following data: 4, 7, 8, 10, 12, 15, 18.

Solution:

Step 1: Find the range:

Range = Maximum − Minimum

= 18 − 4 = 14

Step 2: Find the quartiles:

Q1 = 7 (25th percentile)
Q3 = 15 (75th percentile)

Step 3: Calculate the IQR:

IQR = Q3 - Q1 = 15 - 7 = 8

Answer:

Range: 14
IQR: 8

Summary

Distributions describe how data is spread, focusing on shape, centre, and spread.
Key shapes: symmetrical, skewed (left or right), uniform, bimodal.
Measures of centre: mean, median, mode.
Measures of spread: range, interquartile range (IQR), standard deviation.
Use graphical tools like histograms and box plots to visualise distributions.

Describing Distributions Simplified Revision Notes for Leaving Cert Mathematics