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Door 11 · Grade 7

📊 Statistics (Measures of Center & Variability)

Mean is sensitive to outliers (extreme values)

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Mean, Median, Mode, Range

Measures of Center describe the typical value in a dataset.
MeasureDefinitionExample: 2,4,4,6,9MeanSum divided by count(2+4+4+6+9)=25 / 5 = 5MedianMiddle value when data is sortedSorted: 2,4,4,6,9 — Middle = 4ModeMost frequently occurring value4 appears twice — Mode = 4RangeMaximum minus minimum9 - 2 = 7When dataset has even count: Median = average of the two middle values.
Outliers affect the mean significantly but not the median.

Worked example

Problem: Find mean, median, mode, range: 7, 3, 8, 3, 9, 6

  1. Sorted: 3,3,6,7,8,9.
  2. Mean=36/6=6.
  3. Median=(6+7)/2=6.5.
  4. Mode=3 (appears twice).
  5. Range=9-3=6.

Answer: Range=9-3=6.

Mean Absolute Deviation (MAD)

Mean Absolute Deviation (MAD) measures how spread out the data is — the average distance each data point is from the mean.
Steps to calculate MAD:
• Find the mean of the dataset
• Find each data point's absolute distance from the mean: |value - mean|
• Find the mean of those distances

A smaller MAD means data is clustered near the mean (consistent). A larger MAD means data is spread out.

Worked example

Problem: Find MAD for 10, 12, 14, 16.

  1. Mean = 13.
  2. Distances: |10-13|=3, |12-13|=1, |14-13|=1, |16-13|=3.
  3. Sum=8.
  4. MAD=8/4=2.

Answer: MAD=8/4=2.

Box Plots — Five-Number Summary

A box plot displays data distribution using five key values:
ValueMeaningMinimumSmallest value in the datasetQ1First quartile — median of the lower half (25th percentile)Median (Q2)Middle value (50th percentile)Q3Third quartile — median of the upper half (75th percentile)MaximumLargest value in the datasetInterquartile Range (IQR) = Q3 - Q1. This represents the spread of the middle 50% of data and is resistant to outliers.

Worked example

Problem: Find five-number summary for 5, 7, 8, 12, 15, 18, 21.

  1. Min=5, Q1=7 (lower half 5,7,8 -> median=7), Median=12, Q3=18 (upper half 15,18,21 -> median=18), Max=21.
  2. IQR=18-7=11.

Answer: IQR=18-7=11.

Random Sampling & Population Inferences

Population: The entire group being studied.
Sample: A subset of the population used to make inferences about the whole.
Random sampling gives every member of the population an equal chance of being selected — this minimizes bias and makes the sample representative.
Sample TypeBiased?ExampleRandom from full rosterNo (good)50 students chosen by lotteryFirst 50 to enter cafeteriaYes (biased)Skews towards early arriversFriends or acquaintancesYes (biased)Similar opinions/backgroundsInference: Using sample results to estimate population characteristics. Larger samples give more accurate inferences.

Worked example

Problem: A school has 800 students. You want to know the favorite lunch. Which sample is best? (A) First 50 in cafeteria (B) 50 random from roster (C) 50 friends.

  1. B is best — randomly selected from the complete roster, so every student has equal chance of being chosen.

Answer: B is best — randomly selected from the complete roster, so every student has equal chance of being chosen.