Statistics: measures of dispersion — KCSE Mathematics

In statistics, variance and standard deviation are measures of dispersion that indicate how spread out the data points are in a dataset.

To calculate the variance (C3²) for ungrouped data, follow these steps:

1. Find the mean (average) of the data set.
2. Subtract the mean from each data point and square the result.
3. Sum all the squared results.
4. Divide by the number of data points (n) for population variance or by (n-1) for sample variance.

The standard deviation (C3) is simply the square root of the variance.

Example: Given the data set: 4, 8, 6, 5, 3

1. Mean = (4 + 8 + 6 + 5 + 3) / 5 = 5.2
2. Squared deviations: (4-5.2)² = 1.44, (8-5.2)² = 7.84, (6-5.2)² = 0.64, (5-5.2)² = 0.04, (3-5.2)² = 4.84
3. Sum = 1.44 + 7.84 + 0.64 + 0.04 + 4.84 = 14.8
4. Variance = 14.8 / 5 = 2.96; Standard deviation = √2.96 ≈ 1.72.

This means the data points are spread out around the mean of 5.2.

A box-and-whisker plot is a graphical representation of a dataset's five-number summary, which includes:

• Minimum (Min): The smallest value in the dataset.
• First Quartile (Q₁): The median of the lower half of the data.
• Median: The middle value of the dataset.
• Third Quartile (Q₃): The median of the upper half of the data.
• Maximum (Max): The largest value in the dataset.

To construct a box-and-whisker plot:

1. Draw a number line that accommodates the range of your data.
2. Plot the five-number summary points on the number line.
3. Draw a box from Q₁ to Q₃, with a line at the median.
4. Extend whiskers from the box to the min and max values.

Interpreting the shape:

• If the median is closer to Q₁, the data is positively skewed.
• If the median is closer to Q₃, the data is negatively skewed.
• If the box is symmetrical, the data is normally distributed.

Measures of dispersion are crucial in statistics as they describe the spread of data points in a dataset. The main measures of dispersion include:

• Range: The difference between the highest and lowest values. It provides a simple measure of variability.
• Variance: The average of the squared differences from the mean. It indicates how far the data points are spread out from the mean.
• Standard Deviation: The square root of the variance. It shows how much the individual data points deviate from the mean on average.

To calculate these measures, follow these steps:

1. Range: Subtract the smallest value from the largest value.
2. Variance:
   • Find the mean of the dataset.
   • Subtract the mean from each data point, square the result, and then average those squared results.
3. Standard Deviation: Take the square root of the variance.

Understanding these measures helps in interpreting data effectively and comparing different datasets.