8 – Descriptive Statistics: Measures of Central Tendency

Introduction

Descriptive statistics is like the “who, what, where, and when” of data analysis. It provides simple summaries and graphics that help us understand and make informed decisions about the information we have. When we look at the data, we can use summary statistics to see what’s similar or different among the different parts of the data. This information is vital for understanding things and deciding what to do next!

8.1 Measures of Central Tendency

Measures of central tendency are statistical tools used to determine a dataset’s centre point or typical value. The main measures are the mean, median, and mode. These measures help to summarize and simplify data, making it easier to understand and interpret.

Mean

The mean, or average, is calculated by summing all the values in a dataset and dividing by the number of values. It is the most commonly used measure of central tendency and provides an excellent overall measure for normally distributed data.

Median

The median is the middle value in an ordered dataset. If the dataset has an odd number of values, it is the exact middle value. If it has an even number of values, it is the average of the two middle values. The median is helpful for skewed distributions or when there are outliers.

Mode

The mode is the value that appears most frequently in a dataset. A dataset can have one mode, more than one mode, or no mode. The mode is handy for categorical data, where we want to know which category is the most common.

When to Use Each Measure

• Mean: Best for data without outliers and when all values are equally important.
• Median: Best for skewed data or when outliers are present, as extreme values do not affect it.
• Mode: Best for categorical data to identify the most common category.

Understanding the mean, median, and mode helps summarise and interpret data effectively. Each measure has its strengths and is best used in different situations depending on the nature of the data.

Functions of an Average:

1. Simplifies Data

Provides a single value representing the entire dataset, making complex data more accessible.

2. Facilitates Comparison

Allows quick comparisons between different datasets or groups by providing a standard measure.

3. Identifies Trends

It helps identify trends and patterns over time or across different datasets.

4. Supports Decision Making

Assists in making informed decisions by summarizing data into a meaningful value.

5. Measures Central Tendency

Indicates the central point of a dataset, giving insight into the typical or average value.

6. Balances Out Variability

Averages the variability in data, providing a stable measure less affected by individual data points.

7. Useful in Statistical Analysis

Forms the basis for further statistical analysis, including variance, standard deviation, and hypothesis testing.

8. Enhances Data Presentation

It makes data presentation more transparent and understandable, especially in charts and graphs.

9. Aids in Predictive Analytics

Used in predictive models to estimate future values based on historical data.

10. Supports Quality Control

In quality control processes, averages help monitor and maintain standards.

11. Economic and Social Indicators

It is commonly used in economic and social statistics, such as calculating GDP, unemployment rates, and average incomes.

12. Benchmarking and Performance Evaluation

Used to set benchmarks and evaluate performance against industry standards or historical performance.

Characteristics of a good Average:

1. Simplicity

Easy to understand and calculate.

2. Representativeness

Accurately reflects the central tendency of the dataset.

3. Stability

Not unduly affected by extreme values or outliers.

4. Comprehensiveness

Utilizes all the data points in the dataset.

5. Consistency

Produces similar results when calculated from different samples of the same population.

6. Mathematical Properties

Suitable for further statistical analysis and mathematical operations.

7. Applicability

Relevant and meaningful for the specific context or field of study.

8. Unbiasedness

Not influenced by the distribution shape of the dataset.

9. Clarity

Defines what is typical or average in the dataset.

10. Reproducibility

Can be consistently reproduced by different researchers or analysts.

11. Ease of Interpretation

Results are straightforward to interpret and communicate to others.

12. Inclusiveness

Takes into account all aspects of the data distribution.

13. Relevance

Provides meaningful information that can be used for decision-making.

14. Sensitivity

It is sensitive enough to detect small changes in the dataset.

15. Robustness

Remains reliable and accurate under varying conditions and datasets.

8.2  Various Measures of Average

Measures of average, also known as measures of central tendency, are statistical tools used to summarize data by identifying the central point within that dataset. The main measures of average include the mean, median, and mode. Each measure has unique properties and applications, making them useful in different contexts. This guide will explore these measures in detail, covering their definitions, calculations, advantages, and appropriate use cases.

I) Mean

The mean is the sum of all the values in a dataset divided by the number of values. It is the most commonly used measure of average and provides a comprehensive summary of the data.

Types of Mean

1. Arithmetic Mean

The Arithmetic Mean, often referred to as the average, is a common method for finding the central value of a data set. It is calculated by summing all the values in the dataset and then dividing that total by the number of values.

Formula:

Numerical Example:

Let’s say we have a set of exam scores: 85, 90, 78, 92, and 88.

Arithmetic Mean=85+90+78+92+88 / 5

= 433 / 5

=86.6

So, the Arithmetic Mean of these exam scores is 86.6.

Application:

The Arithmetic Mean is widely used in areas like economics, finance, and daily activities. It provides an easy way to understand the central tendency of data. For instance, you can use it to find the average income of a group or the average temperature over a period of time. It’s a simple and effective tool for analyzing data in many contexts.

2. Weighted Mean

The Weighted Mean differs from the Arithmetic Mean by giving different values in a dataset varying levels of importance, or “weights,” depending on their significance.

Formula:

Numerical Example:

Imagine a student’s grades in three subjects:

• Math (40% weight),
• Science (30% weight),
• English (30% weight),

with respective scores of 85, 90, and 88.

Weighted Mean=(0.4×85)+(0.3×90)+(0.3×88) / 0.4+0.3+0.3​

Calculating:

Weighted Mean=34+27+26.4 / 1 = 87.4

Thus, the Weighted Mean of the grades is 87.4.

Application:

The Weighted Mean is particularly useful when different values contribute unequally to the final result. For instance, in a stock portfolio, different investments may have varying amounts of capital allocated to them. The Weighted Mean helps in calculating the portfolio’s overall return. Similarly, in education, courses with more credits have greater influence on a student’s GPA, and using the Weighted Mean reflects this accurately.

3. Geometric Mean

The Geometric Mean is a type of average used for data sets involving multiplicative relationships, such as growth rates. It is particularly helpful when values are multiplied together or when dealing with percentages and ratios.

Formula:

Numerical Example:

Let’s say an investment grows by 10%, 50%, and then declines by 20% over three years. The corresponding growth factors are 1.10, 1.50, and 0.80.

Geometric Mean=(1.10×1.50×0.80)1/3​

Calculating:

Geometric Mean=(1.32)1/3≈1.097 or 9.7%

Thus, the Geometric Mean of the growth rates is approximately 9.7%.

Application:

The Geometric Mean is frequently used in fields like finance and economics, especially for calculating average growth rates over time. It accounts for the compounding effect, making it suitable for determining the average return on an investment over multiple periods. It is also useful in averaging ratios or percentages where the relationships between values are multiplicative, not additive.

4. Harmonic Mean

The Harmonic Mean is another average useful when dealing with rates or ratios, especially when the data points are highly variable.

Formula:

Numerical Example:

Suppose a car travels the same distance at different speeds: 60 km/h, 80 km/h, and 100 km/h.

First, calculate the sum of the reciprocals:

1 / 60=0.01667, 1 / 80=0.0125, 1 / 100=0.01

Now, sum these values:

0.01667+0.0125+0.01=0.03917

Finally, calculate the Harmonic Mean:

So, the Harmonic Mean of the speeds is approximately 76.6 km/h.

Application:

The harmonic mean is particularly effective when the average rate is needed. It is commonly used in calculating average speeds, where the total distance is divided by the total time. Another application is in finance, where the Harmonic Mean can be used to calculate the average price-earnings ratio (P/E ratio) of a portfolio of stocks.

Advantages of the Mean

• Comprehensive: Utilizes all data points.
• Mathematically Rigorous: Suitable for further statistical analysis.
• Easy to Understand: Simple calculation and interpretation.

Disadvantages of the Mean

• Sensitive to Outliers: Can be heavily affected by extreme values.
• Not Always Representative: This may not accurately reflect the central tendency in skewed distributions.

II) Median

The Median is the middle value in a dataset when the numbers are arranged in either ascending or descending order. If the dataset has an odd number of values, the median is the exact middle value. If the dataset has an even number of values, the median is the average of the two middle values.

Formula:

For a dataset sorted in ascending order:

Numerical Example:

Consider the following exam scores: 72, 85, 90, 93, 95.

• Step 1: Arrange the scores in ascending order: 72, 85, 90, 93, 95.
• Step 2: Since n=5 (odd), the median is the middle value:

Median=90

Now, let’s look at an example with an even number of values: 72, 85, 90, 93.

• Step 1: Arrange the scores in ascending order: 72, 85, 90, 93.
• Step 2: Since n=4 (even), the median is the average of the two middle values:

Median=85+90 / 2 = 87.5

Application:

The Median is particularly useful for datasets with skewed distributions, where extreme values can distort the mean. For example, when analyzing household incomes, the median provides a more accurate representation of the typical income, as it is not affected by very high or very low values.

Advantages of the Median:

• Resistant to outliers: Not influenced by extreme values.
• Simple to interpret: Represents the central value clearly.

Disadvantages of the Median:

• Ignores some data: Only considers the middle value(s), missing out on other data points.
• Less precise for large datasets: May not capture finer variations in larger datasets.

III) Mode

The Mode is the value that appears most frequently in a dataset. A dataset can have more than one mode if multiple values occur with the same highest frequency or no mode if all values occur with the same frequency.

Formula:

There is no specific formula for calculating the mode, as it is simply the value that occurs most frequently in the dataset.

Numerical Example:

Consider the following dataset of test scores: 85, 90, 75, 90, 80, 90, 85.

• Arranging the scores: 75, 80, 85, 85, 90, 90, 90.

The Mode is the most frequent value:

Mode=90

If we have another dataset where the scores are 75, 80, 85, 90:

• All values occur only once.

In this case, the dataset has no mode because no value repeats.

Application:

The Mode is beneficial in categorical data, where we are interested in the most common category. For example, understanding the most frequently purchased product size or colour in retail can help optimize inventory management.

Advantages of the Mode

• Applicable to Categorical Data: Useful for identifying the most common category.
• Unaffected by Extreme Values: Not influenced by outliers.

Disadvantages of the Mode

• May Not Be Unique: Multiple modes can complicate interpretation.
• Not Useful for Small Datasets: This may not provide meaningful information in small datasets.

Comparison of Measures of Average

When to Use Each Measure

• Mean: Best for customarily distributed data without outliers.
• Median: Best for skewed data or when outliers are present.
• Mode: Best for categorical data to identify the most common category.

Advantages and Disadvantages

• Mean: Sensitive to outliers but provides a comprehensive measure.
• Median: Robust to outliers but does not utilize all data points.
• Mode: Useful for categorical data but may not always exist or be unique.

Applications of Measures of Average

Education

• Mean: Used to calculate average test scores.
• Median: Used to understand the middle-performance level.
• Mode: Identifies the most common grade.

Healthcare

• Mean: Analyzes average patient data such as blood pressure.
• Median: Determines the median survival time in clinical studies.
• Mode: Identifies the most common symptom.

Economics

• Mean: Calculates average income and expenditure.
• Median: Assesses median household income to understand economic status.
• Mode: Identifies the most common price point in a market.

Social Sciences

• Mean: Summarizes survey results.
• Median: Determines the median age in a population study.
• Mode: Identifies the most common response in a survey.

Sports

• Mean: Calculates average player performance.
• Median: Assesses the median score of games.
• Mode: Identifies the most frequent score.

Understanding and correctly applying the measures of average—mean, median, and mode—are fundamental for accurate data analysis and interpretation. Each measure has its unique properties and appropriate contexts of use. By recognizing their advantages and limitations, one can choose the most suitable measure to accurately represent and summarize data, leading to more informed decisions and insights.

Practical Problems with Mean, Median, and Mode

Problem Statement

Consider the following dataset representing the scores of 10 students in a math test:

Student	Score
A	55
B	67
C	45
D	70
E	68
F	80
G	90
H	85
I	75
J	60

We will calculate the mean, median, and mode of the scores.

Solution

Step 1: Organize the Data

First, we list the scores in ascending order for ease of calculation.

Student	Score
C	45
A	55
J	60
B	67
E	68
D	70
I	75
F	80
H	85
G	90

Step 2: Calculate the Mean

Formula:

​​

Calculation:

Mean=45+55+60+67+68+70+75+80+85+90 / 10

=695 / 10

=69.5

Mean Score: 69.5

Step 3: Calculate the Median

Method: For an even number of observations, the median is the average of the two middle numbers.

Calculation: The middle scores are the 5th and 6th values in the ordered list: 68 and 70.

Median=68+70 / 2

=138 / 2

=69

Median Score: 69

Step 4: Calculate the Mode

Method: The mode is the most frequently occurring value in the dataset.

Calculation: Each score occurs exactly once, so this dataset has no mode.

Mode: None

Conclusion

• Mean: The average score of the students is 69.5.
• Median: The middle score is 69.
• Mode: There is no mode since no score is repeated.

This example demonstrates how to calculate and interpret a dataset’s mean, median, and mode, helping to understand the data’s central tendency and distribution.

Relationship Between Mean, Median, and Mode

Understanding the relationship between mean, median, and mode is essential for interpreting data distributions effectively. These measures of central tendency often provide different insights about the dataset, and their relationship can give us information about the distribution’s skewness.

General Relationship

For a symmetrical distribution (normal distribution):

• Mean = Median = Mode

For a skewed distribution:

• The position of the mean, median, and mode relative to each other helps identify the skewness of the dataset.

Skewness and Central Tendency

1. Symmetrical Distribution (Normal Distribution)

• In a perfectly symmetrical distribution, the mean, median, and mode are all equal and located at the center of the distribution.
• Example:
  

2. Positively Skewed Distribution (Right-Skewed)

• • The mean is greater than the median and greater than the mode.
  • The long tail is on the right side.
  • Example: Income distribution, where many high values (outliers) pull the mean to the right.
  • Relationship: Mean > Median > Mode
  • Diagram:
    

    3. Negatively Skewed Distribution (Left-Skewed)
    • The mean is less than the median, which is less than the mode.
    • The long tail is on the left side.
    • Example: Age at retirement, where a few early retirements pull the mean to the left.
    • Relationship: Mode > Median > Mean
    • Diagram:
      

      


Mathematical Relationship

In a moderately skewed distribution, the following empirical relationship often holds:

Mode≈3×Median−2×Mean

This formula, known as the empirical relationship or the Pearson mode skewness, provides a way to estimate the mode if the mean and median are known.

Practical Example

Consider a dataset with the following values representing students’ test scores: [40, 50, 60, 70, 80, 90, 100].

Mean:
Mean=40+50+60+70+80+90+100 / 7

=490 / 7

=70

Median: Since the number of values is odd, the median is the middle value.
Median=70

Mode: All values occur only once, so there is no mode.

Since the mean and median are equal and there is no mode, this dataset is an example of a symmetrical distribution.

Understanding the relationship between mean, median, and mode helps in identifying the nature of the data distribution:

• Symmetrical Distribution: Mean = Median = Mode.
• Positively Skewed Distribution: Mean > Median > Mode.
• Negatively Skewed Distribution: Mode > Median > Mean.

These relationships allow us to gain insights into the data’s distribution and the presence of any skewness, which is crucial for accurate data analysis and interpretation.

8.3      Measures of Dispersion

Measures of dispersion, also known as variability or spread, are statistical tools used to quantify the extent to which individual data points differ from the central tendency of a dataset. While measures of central tendency (such as mean, median, and mode) provide information about a dataset’s typical or average value, measures of dispersion provide insights into the variability, diversity, or spread of the data points around the central value.

Importance of Measures of Dispersion

Understanding the dispersion of data is crucial for several reasons:

1. Understanding Variability: It helps understand how much the data points deviate from the average, providing a clearer picture of the dataset’s distribution.
2. Assessing Data Quality: High dispersion indicates more significant variability among data points, which might imply less consistency or reliability in the dataset.
3. Making Inferences: Measures of dispersion are essential for making accurate inferences and predictions based on the dataset.
4. Comparing Datasets: It allows for comparing the variability of different datasets, aiding in identifying patterns, trends, or anomalies.

Common Measures of Dispersion

1. Range: The most straightforward measure of dispersion, calculated as the difference between the maximum and minimum values in the dataset. While easy to compute, it is susceptible to outliers and may not provide a robust measure of variability.
2. Variance: The average of the squared differences from the mean. It quantifies the spread of data points around the mean. However, variance is not in the same units as the original data, making interpretation challenging.
3. Standard Deviation: The square root of the variance, indicating the average distance of data points from the mean. Standard deviation is in the same units as the original data, making it more interpretable than variance.
4. Mean Absolute Deviation (MAD): The average of the absolute differences between each data point and the mean. MAD is less sensitive to extreme values compared to variance and standard deviation.

Application of Measures of Dispersion

1. Finance: Assessing investment risks by measuring the variability of returns.
2. Manufacturing: Monitoring the consistency of product quality.
3. Education: Evaluating the variability of test scores among students.
4. Healthcare: Analyzing the variability of patient outcomes in clinical trials.
5. Economics: Understanding the variability of economic indicators such as GDP and inflation rates.

Measures of dispersion complement measures of central tendency by providing valuable insights into the variability and spread of data points in a dataset. By quantifying the extent of variability, these measures enable better understanding, analysis, and interpretation of data, facilitating informed decision-making and drawing reliable conclusions from statistical studies.

Common Measures of Dispersion

Range

• Definition: The range is the simplest measure of dispersion, calculated as the difference between the maximum and minimum values in the dataset.
• Calculation:
  Range=Maximum Value−Minimum Value
• Advantages: Easy to compute and understand.
• Disadvantages: Highly sensitive to outliers and may not provide a robust measure of variability.

Variance

• Definition: Variance is the average of the squared differences from the mean. It quantifies the spread of data points around the mean.
• Calculation:
  ​

• Advantages: Provides a precise measure of dispersion.
• Disadvantages: Not in the same units as the original data, making interpretation challenging.

Standard Deviation

• Definition: The square root of the variance, indicating the average distance of data points from the mean. Standard deviation is in the same units as the original data, making it more interpretable than variance.
• Calculation:
  ​
• Advantages: Provides a measure of dispersion in the same units as the original data.
• Disadvantages: Still sensitive to outliers, although less so than the range.

Mean Absolute Deviation (MAD)

• Definition: The average absolute differences between each data point and the mean. MAD is less sensitive to extreme values compared to variance and standard deviation.
• Calculation:
  ​
• Advantages: Less sensitive to outliers compared to variance and standard deviation.
• Disadvantages: Slightly more complex to compute than the range.

Conclusion

Measures of dispersion provide valuable insights into the spread or variability of data points in a dataset. While each measure has advantages and disadvantages, understanding its characteristics and graphical representations can aid in selecting the most appropriate measure for a given dataset and analytical purpose.