Central Tendency: Median and Mode

• Measures of central tendency and variability are essential for understanding and summarizing data
• These measures provide insights into the typical values and spread within a dataset

Median

• The median is the middle value in a dataset when the data is ordered from least to greatest
• To find the median, first sort the data
• If there is an odd number of data points, the median is the middle value
• If there is an even number of data points, the median is the average of the two middle values
• The median is resistant to outliers, making it a robust measure of central tendency
• It represents the 50th percentile of the data

Mode

• The mode is the value that appears most frequently in a dataset
• A dataset can have one mode (unimodal), more than one mode (bimodal or multimodal), or no mode if all values occur with the same frequency
• The mode is useful for identifying the most common category or value in a dataset
• Unlike the mean and median, the mode can be used with nominal (categorical) data

Range

• The range is the difference between the maximum and minimum values in a dataset
• It provides a simple measure of the total spread of the data
• Range = Maximum Value – Minimum Value
• The range is sensitive to outliers because it only considers the extreme values

Interquartile Range (IQR)

• The interquartile range (IQR) is a measure of statistical dispersion, representing the range of the middle 50% of the data
• It is calculated as the difference between the third quartile (Q3) and the first quartile (Q1)
• IQR = Q3 – Q1
• The IQR is less sensitive to outliers than the range because it focuses on the central portion of the data

Quartiles: Q1 and Q3

• Quartiles divide a dataset into four equal parts
• The first quartile (Q1) is the median of the lower half of the data
• It represents the 25th percentile, meaning 25% of the data falls below this value
• The third quartile (Q3) is the median of the upper half of the data
• It represents the 75th percentile, meaning 75% of the data falls below this value
• To find Q1 and Q3:
  • Sort the data from least to greatest
  • Find the median of the entire dataset (Q2)
  • Find the median of the data points below Q2 (Q1)
  • Find the median of the data points above Q2 (Q3)

Data

• Data are the individual pieces of information or observations collected and analyzed
• Data can be numerical (quantitative) or categorical (qualitative)
• Understanding the type and distribution of data is crucial for choosing appropriate statistical measures
• Data is often organized in datasets, where each row represents an observation, and each column represents a variable
• Before analysis, data should be cleaned and preprocessed to handle missing values, outliers, and inconsistencies

Box Plot

• A box plot (also known as a box-and-whisker plot) is a graphical representation of data that displays the summary of the data
• A box plot displays the five-number summary: minimum, Q1, median (Q2), Q3, and maximum
• Box plots provide a visual way to assess the center, spread, and skewness of a dataset
• The box in a box plot represents the IQR, with the median marked inside the box
• Whiskers extend from the box to the minimum and maximum values within 1.5 times the IQR from the quartiles
• Values outside this range are considered outliers and are plotted as individual points
• Box plots are useful for comparing the distributions of multiple datasets
• They can quickly reveal differences in medians, spreads, and the presence of outliers
• Box plots can be drawn horizontally or vertically
• The length of the box indicates the spread of the data, with a longer box indicating greater variability
• The position of the median within the box provides insight into the skewness of the data
• If the median is closer to Q1, the data is skewed to the right (positively skewed)
• If the median is closer to Q3, the data is skewed to the left (negatively skewed)