Calculating Measures of Central Tendency

Oct 13, 2025 by wbrum.org 8 views

Summarizing Data (Measures of Central Tendency)

A group of 10 young professionals living in Three Rivers were surveyed about their monthly expenses. The data collected includes their spending on five categories: Rent, Utilities, Groceries, Entertainment, and Transportation. The objective of this analysis is to understand the measures of central tendency, which are essential in summarizing and interpreting the data.

What are Measures of Central Tendency?

Measures of central tendency are statistical tools used to describe the central or typical value of a dataset. They provide a single value that best represents the entire dataset. The three primary measures of central tendency are:

Mean: The average value of a dataset, calculated by summing up all the values and dividing by the number of values.
Median: The middle value of a dataset when it is arranged in ascending or descending order. If the dataset has an even number of values, the median is the average of the two middle values.
Mode: The most frequently occurring value in a dataset.

Calculating Measures of Central Tendency

To calculate the measures of central tendency, we need to first collect and organize the data. Let's assume the data collected from the survey is as follows:

Category	Rent	Utilities	Groceries	Entertainment	Transportation
1	800	100	300	200	150
2	900	120	350	250	180
3	700	90	250	150	120
4	600	80	200	100	90
5	500	70	150	80	60
6	400	60	100	60	40
7	300	50	50	40	30
8	200	40	0	20	20
9	100	30	0	10	10
10	0	20	0	0	0

Calculating the Mean

To calculate the mean, we need to sum up all the values and divide by the number of values.

import numpy as np
data = np.array([
[800, 100, 300, 200, 150],
[900, 120, 350, 250, 180],
[700, 90, 250, 150, 120],
[600, 80, 200, 100, 90],
[500, 70, 150, 80, 60],
[400, 60, 100, 60, 40],
[300, 50, 50, 40, 30],
[200, 40, 0, 20, 20],
[100, 30, 0, 10, 10],
[0, 20, 0, 0, 0]
])
mean = np.mean(data, axis=0)
print("Mean:", mean)

Calculating the Median

To calculate the median, we need to arrange the data in ascending or descending order and find the middle value.

import numpy as np
data = np.array([
[800, 100, 300, 200, 150],
[900, 120, 350, 250, 180],
[700, 90, 250, 150, 120],
[600, 80, 200, 100, 90],
[500, 70, 150, 80, 60],
[400, 60, 100, 60, 40],
[300, 50, 50, 40, 30],
[200, 40, 0, 20, 20],
[100, 30, 0, 10, 10],
[0, 20, 0, 0, 0]
])
median = np.median(data, axis=0)
print("Median:", median)

Calculating the Mode

To calculate the mode, we need to find the most frequently occurring value in the dataset.

import numpy as np
data = np.array([
[800, 100, 300, 200, 150],
[900, 120, 350, 250, 180],
[700, 90, 250, 150, 120],
[600, 80, 200, 100, 90],
[500, 70, 150, 80, 60],
[400, 60, 100, 60, 40],
[300, 50, 50, 40, 30],
[200, 40, 0, 20, 20],
[100, 30, 0, 10, 10],
[0, 20, 0, 0, 0]
])
mode = np.bincount(data.flat).argmax()
print("Mode:", mode)

Interpreting the Results

The measures of central tendency provide a summary of the dataset. The mean, median, and mode can be used to understand the central tendency of the data.

Mean: The mean is the average value of the dataset. It is sensitive to extreme values and may not accurately represent the data if there are outliers.
Median: The median is the middle value of the dataset. It is a better representation of the data than the mean if there are outliers.
Mode: The mode is the most frequently occurring value in the dataset. It is a useful measure of central tendency if the data is categorical.

Conclusion

Measures of central tendency are essential in summarizing and interpreting data. The mean, median, and mode provide a summary of the dataset and can be used to understand the central tendency of the data. By calculating and interpreting these measures, we can gain insights into the data and make informed decisions.

References

Khan Academy. (n.d.). Measures of Central Tendency. Retrieved from https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/mean-median-mode/v/measures-of-central-tendency
Wikipedia. (n.d.). Measures of Central Tendency. Retrieved from https://en.wikipedia.org/wiki/Measures_of_central_tendency

Future Work

In the future, we can explore other measures of central tendency, such as the geometric mean and the harmonic mean. We can also investigate the use of measures of central tendency in real-world applications, such as finance and economics.

Code

The code used in this analysis is available in the following Python script:

import numpy as np
data = np.array([
[800, 100, 300, 200, 150],
[900, 120, 350, 250, 180],
[700, 90, 250, 150, 120],
[600, 80, 200, 100, 90],
[500, 70, 150, 80, 60],
[400, 60, 100, 60, 40],
[300, 50, 50, 40, 30],
[200, 40, 0, 20, 20],
[100, 30, 0, 10, 10],
[0, 20, 0, 0, 0]
])
mean = np.mean(data, axis=0)
print("Mean:", mean)
median = np.median(data, axis=0)
print("Median:", median)
mode = np.bincount(data.flat).argmax()
print("Mode:", mode)

Q&A: Measures of Central Tendency

In our previous article, we discussed the measures of central tendency, which are essential in summarizing and interpreting data. In this article, we will answer some frequently asked questions about measures of central tendency.

Q: What is the difference between the mean and the median?

A: The mean and the median are both measures of central tendency, but they are calculated differently. The mean is the average value of a dataset, calculated by summing up all the values and dividing by the number of values. The median, on the other hand, is the middle value of a dataset when it is arranged in ascending or descending order.

Q: When should I use the mean and when should I use the median?

A: You should use the mean when the data is normally distributed and there are no outliers. However, if the data is skewed or has outliers, you should use the median. The median is a better representation of the data in these cases.

Q: What is the mode and how is it calculated?

A: The mode is the most frequently occurring value in a dataset. It is calculated by finding the value that appears most frequently in the dataset.

Q: Can a dataset have more than one mode?

A: Yes, a dataset can have more than one mode. This occurs when there are multiple values that appear with the same frequency, and no single value appears more frequently than the others.

Q: What is the geometric mean and how is it calculated?

A: The geometric mean is a measure of central tendency that is used for datasets that are skewed or have outliers. It is calculated by taking the nth root of the product of the values, where n is the number of values.

Q: What is the harmonic mean and how is it calculated?

A: The harmonic mean is a measure of central tendency that is used for datasets that are skewed or have outliers. It is calculated by taking the reciprocal of the average of the reciprocals of the values.

Q: Can I use the measures of central tendency for categorical data?

A: No, the measures of central tendency are used for numerical data, not categorical data. For categorical data, you should use other measures of central tendency, such as the mode.

Q: How do I choose the right measure of central tendency for my data?

A: You should choose the measure of central tendency that best represents your data. If your data is normally distributed and has no outliers, you can use the mean. If your data is skewed or has outliers, you should use the median or the mode.

Q: Can I use the measures of central tendency for time series data?

A: Yes, you can use the measures of central tendency for time series data. However, you should be careful when using the mean, as it can be affected by outliers and trends in the data.

Q: How do I interpret the results of the measures of central tendency?

A: You should interpret the results of the measures of central tendency in the context of your data. For example, if you are analyzing the average salary of a group of people, you should consider the median salary as well as the mean salary.

Conclusion

Measures of central tendency are essential in summarizing and interpreting data. By understanding the different types of measures of central tendency and how to calculate them, you can make informed decisions about your data. Remember to choose the right measure of central tendency for your data and to interpret the results in the context of your data.

References

Khan Academy. (n.d.). Measures of Central Tendency. Retrieved from https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/mean-median-mode/v/measures-of-central-tendency
Wikipedia. (n.d.). Measures of Central Tendency. Retrieved from https://en.wikipedia.org/wiki/Measures_of_central_tendency

Future Work

In the future, we can explore other measures of central tendency, such as the trimmed mean and the Winsorized mean. We can also investigate the use of measures of central tendency in real-world applications, such as finance and economics.

Code

The code used in this analysis is available in the following Python script:

import numpy as np
data = np.array([
[800, 100, 300, 200, 150],
[900, 120, 350, 250, 180],
[700, 90, 250, 150, 120],
[600, 80, 200, 100, 90],
[500, 70, 150, 80, 60],
[400, 60, 100, 60, 40],
[300, 50, 50, 40, 30],
[200, 40, 0, 20, 20],
[100, 30, 0, 10, 10],
[0, 20, 0, 0, 0]
])
mean = np.mean(data, axis=0)
print("Mean:", mean)
median = np.median(data, axis=0)
print("Median:", median)
mode = np.bincount(data.flat).argmax()
print("Mode:", mode)

This script calculates the mean, median, and mode of the dataset using the NumPy library.

Category	Rent	Utilities	Groceries	Entertainment	Transportation
1	800	100	300	200	150
2	900	120	350	250	180
3	700	90	250	150	120
4	600	80	200	100	90
5	500	70	150	80	60
6	400	60	100	60	40
7	300	50	50	40	30
8	200	40	0	20	20
9	100	30	0	10	10
10	0	20	0	0	0

Category	Rent	Utilities	Groceries	Entertainment	Transportation
1	800	100	300	200	150
2	900	120	350	250	180
3	700	90	250	150	120
4	600	80	200	100	90
5	500	70	150	80	60
6	400	60	100	60	40
7	300	50	50	40	30
8	200	40	0	20	20
9	100	30	0	10	10
10	0	20	0	0	0

Category	Rent	Utilities	Groceries	Entertainment	Transportation
1	800	100	300	200	150
2	900	120	350	250	180
3	700	90	250	150	120
4	600	80	200	100	90
5	500	70	150	80	60
6	400	60	100	60	40
7	300	50	50	40	30
8	200	40	0	20	20
9	100	30	0	10	10
10	0	20	0	0	0