Data Analysis
The Final Inspection
Data Analysis is the final inspection where you interpret all the various components to ensure everything fits together accurately. This section covers approximately 30% of Quantitative Reasoning questions—the largest portion!
- Descriptive statistics (mean, median, mode, range, standard deviation)
- Data interpretation (graphs, tables, charts)
- Probability and counting methods
- Distributions and data patterns
- Percentiles and quartiles
Measures of Central Tendency
Mean (Average)
Sum of all values divided by the number of values.
Mean = (Sum of all values) / (Number of values)
Median
The middle value when data is arranged in order. If there's an even number of values, it's the average of the two middle values.
Example: For 1, 3, 5, 7, 9 → median is 5. For 2, 4, 6, 8 → median is (4+6)/2 = 5
Mode
The value that appears most frequently. A dataset can have no mode, one mode, or multiple modes.
Range
The difference between the maximum and minimum values.
Range = Maximum - Minimum
Standard Deviation
What is Standard Deviation?
Standard deviation measures how much data values differ from the mean. A larger standard deviation indicates more spread out data; a smaller one indicates data clustered close to the mean.
Example:
Calculate the standard deviation for the data: 0, 7, 8, 10, 10
Solution:
- Find the mean: (0 + 7 + 8 + 10 + 10) / 5 = 35 / 5 = 7
- Find squared differences from mean:
- (0-7)2 = 49
- (7-7)2 = 0
- (8-7)2 = 1
- (10-7)2 = 9
- (10-7)2 = 9
- Average of squared differences: (49 + 0 + 1 + 9 + 9) / 5 = 68 / 5 = 13.6
- Standard deviation = √13.6 ≈ 3.69
- Answer: ≈ 3.7
Key Property:
If all values in a dataset are increased by the same amount, the standard deviation stays the same (it measures spread, not position).
Counting Methods
Combinations: Order Does NOT Matter
Use combinations when selecting items where the order does not matter.
C(n,k) = n! / (k!(n-k)!)
"n choose k" = number of ways to choose k items from n items
Example:
To select a 3-person committee from 9 students, how many ways are there?
Solution:
- C(9,3) = 9! / (3!(6!))
- = (9 × 8 × 7) / (3 × 2 × 1)
- = 504 / 6
- Answer: 84 ways
Permutations: Order DOES Matter
Use permutations when selecting items where the order matters.
P(n,k) = n! / (n-k)!
Example: First, second, and third place in a race of 8 runners = P(8,3) = 8×7×6 = 336
Probability
Basic Probability Formula
P(Event) = (Number of favorable outcomes) / (Total number of outcomes)
Probability always ranges from 0 to 1 (or 0% to 100%)
Independent Events
If two events E and F are independent (one doesn't affect the other), the probability of both occurring is:
P(E and F) = P(E) × P(F)
Example:
If a fair die is rolled twice, what's the probability of rolling a 3 both times?
Solution:
- P(rolling a 3 on first roll) = 1/6
- P(rolling a 3 on second roll) = 1/6
- P(both rolls are 3) = 1/6 × 1/6
- Answer: 1/36
Mutually Exclusive Events
If events cannot happen at the same time (mutually exclusive), the probability of either occurring is:
P(E or F) = P(E) + P(F)
Example: P(rolling a 2 or a 5) = 1/6 + 1/6 = 2/6 = 1/3
Normal Distribution
The Bell Curve
In a perfectly symmetric normal distribution (bell curve), the mean, median, and mode are all equal and located at the center.
The 68-95-99.7 Rule
- About 68% of data falls within 1 standard deviation of the mean
- About 95% of data falls within 2 standard deviations of the mean
- About 99.7% of data falls within 3 standard deviations of the mean
Key Fact to Remember:
About two-thirds (68%) of the data in a normal distribution falls within 1 standard deviation of the mean.
Percentiles and Quartiles
Percentiles
The nth percentile is the value below which n% of the data falls. For example, the 75th percentile means 75% of values are below this point.
Quartiles
- Q1 (First Quartile): 25th percentile
- Q2 (Second Quartile): 50th percentile = Median
- Q3 (Third Quartile): 75th percentile
Interquartile Range (IQR)
The IQR measures the spread of the middle 50% of data:
IQR = Q3 - Q1
Data Interpretation Tips
- ✓Read labels carefully: Check axis labels, units, legends, and titles on all graphs and tables.
- ✓Watch for scale: Graphs can be misleading if axes don't start at zero or use irregular intervals.
- ✓Calculate carefully: Use the on-screen calculator for complex calculations, but double-check your work.
- ✓Estimate when possible: Sometimes you can eliminate wrong answers through estimation without precise calculation.
Ready to Practice Data Analysis?
Master these critical data interpretation and statistical reasoning skills.