What is the fundamental difference between the Mean and the Median in terms of data sensitivity?

The Mean is sensitive to every value in the data set, meaning a single outlier can significantly pull it away from the center. In contrast, the Median is a positional measure that only considers the middle value, making it robust against extreme outliers.

When should the Interquartile Range (IQR) be used instead of the standard Range?

The IQR should be used when a data set contains outliers or extreme values that would artificially inflate the standard Range. Because the IQR only measures the spread of the middle 50% of the data, it provides a more reliable measure of 'typical' variability.

How does the calculation of the mean change when moving from raw data to a frequency table?

For raw data, you simply sum the values and divide by $n$. In a frequency table, you must multiply each value (or midpoint) by its frequency ($fx$), sum those products ($\\sum fx$), and then divide by the total frequency ($\\sum f$).

What is a common mistake when calculating the mean for grouped data in a frequency table?

A common error is using the class width or the lower boundary instead of the class midpoint. Because the specific values within a group are unknown, the midpoint is the only statistically valid representative for that interval.

Why might a student get an impossible value for Variance, and how can they check it?

Variance must always be positive because it is based on squared deviations. If a student calculates a negative variance, they likely swapped the terms in the formula $\\frac{\\sum x^2}{n} - \\bar{x}^2$ or made a sign error during summation.

What error occurs if a student calculates the median of a data set without ordering it first?

The median is defined as the middle value of an *ordered* set. If the data is not sorted from smallest to largest, the 'middle' value of the list is essentially random and carries no statistical meaning regarding the center of the data.

Define 'Standard Deviation' in the context of data spread.

Standard Deviation is the square root of the variance, representing the average distance of data points from the mean. It is preferred over variance because it is expressed in the same units as the original data, making it easier to interpret.

What is a 'Bimodal' distribution?

A bimodal distribution is a data set that has two distinct modes, meaning two different values occur with the same highest frequency. This often suggests that the data set is actually composed of two different underlying groups.

What does the 70th percentile represent in a data set?

The 70th percentile is the value below which 70% of the data falls. Conversely, it implies that 30% of the data values in the set are greater than or equal to this value.

State the computational formula for Variance ($\\sigma^2$).

The formula is $\\sigma^2 = \\frac{\\sum x^2}{n} - \\bar{x}^2$. This is calculated by taking the sum of the squares of all data points divided by the number of points, then subtracting the square of the mean.

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Data Presentation & Interpretation

Basic Statistical Measures

Summary

Statistical measures provide a quantitative summary of data sets, primarily focusing on 'location' (where the center lies) and 'spread' (how much the data varies). Understanding the strengths and limitations of each measure—such as the mean's sensitivity to outliers versus the median's robustness—is essential for accurate data interpretation and selection of appropriate analytical methods.

1. Measures of Central Tendency (Location)

Mean ( $\\bar{x}$ ): The arithmetic average calculated by summing all values and dividing by the total count ( $n$ ). It is the most common measure but is highly sensitive to extreme values (outliers).
Median: The middle value when data is arranged in ascending order. If $n$ is even, it is the midpoint of the two central values; it is preferred for skewed data as it resists outlier influence.
Mode: The most frequently occurring value in a data set. A set can be bimodal (two modes) or have no mode, and it is the only measure applicable to qualitative (categorical) data.

A symmetrical bell curve showing that in a normal distribution, the mean, median, and mode align at the center peak.

2. Measures of Spread (Dispersion)

3. Quartiles and Percentiles

4. Statistical Notation and Formulas

5. Key Distinctions: Mean vs. Median

Feature	Mean	Median
Sensitivity	High (affected by outliers)	Low (robust)
Data Usage	Uses every value in the set	Uses only the middle position(s)
Best Use	Symmetrical data without outliers	Skewed data or data with outliers

Decision Criteria: Use the Mean when you need to account for the total value of the data (e.g., total revenue). Use the Median when you want to represent a 'typical' member of a group where extremes might distort the average (e.g., household income).

6. Exam Strategy & Common Pitfalls