Measures of Central Tendency
When analyzing a dataset, it's helpful to use measures of central tendency to understand the 'center' or typical value within the data. These measures include the mean, median, and mode.
The Mean
The mean is calculated by adding all the values in the dataset and dividing by the total number of values. It's also called the average.
Example:
In a class of 5 students, the test scores are 63, 74, 80, 85, and 91. To find the mean, add them up and divide by 5 (the number of students):
(63 + 74 + 80 + 85 +91)/5 = 393/5 = 78.6
The mean can give a quick summary of the data, but it may not represent all values well, especially if there are extreme values (called outliers) that can skew the mean. An outlier is a value much higher or lower than the rest of the data. The farther an outlier is from the mean, the greater impact it has on it.
Spread, or dispersion, refers to how much the values in a dataset vary or are scattered around the mean. Greater spread indicates more variability in the data, while smaller spread suggests that the data points are closer to the center.
The Median
The median is the middle value in a dataset when the values are arranged in order. If there’s an even number of values, the median is the average of the two middle numbers.
Example:
Using the same test scores {63, 74, 80, 85, 91}, the median is the third score, 80, because it’s in the middle.
If the scores were 5, 74, 80, 85, and 100, the median would still be 80, even though one score is much lower than the others. Unlike the mean, the median is not affected by outliers.
The Mode
The mode of a set of numbers is the value that appears most frequently. In a set where all numbers occur only once, there is no mode.
Example:
In the test scores {63, 74, 80, 85, 91}, since each score appears only once, there is no mode.
Outliers
Outliers can have a big impact on the mean but not on the median or mode. For example, if the test scores {63, 74, 80, 85, 91} include one outlier (changing 63 to 5), the mean drops from 78.6 to 67, but the median (80) remains the same.