Math Problem Statement

Calculate the 85th percentile of the data shown (11, 17.8, 18.9, 20.7, 45.8, 54.7, 56.1, 59.2, 60.2, 69.9).

Solution

To calculate the 85th percentile for the given dataset, we follow these steps:

  1. Arrange the data in ascending order (this step is unnecessary here, as the data is already sorted): 11,17.8,18.9,20.7,45.8,54.7,56.1,59.2,60.2,69.911, 17.8, 18.9, 20.7, 45.8, 54.7, 56.1, 59.2, 60.2, 69.9

  2. Calculate the rank corresponding to the 85th percentile using the formula: Rank=P×(N+1)\text{Rank} = P \times (N + 1) Where:

    • PP is the percentile (0.85 for the 85th percentile),
    • NN is the number of data points.

    In this case, N=10N = 10, so the rank is: Rank=0.85×(10+1)=0.85×11=9.35\text{Rank} = 0.85 \times (10 + 1) = 0.85 \times 11 = 9.35

  3. Interpret the rank: The rank 9.35 means that the 85th percentile lies between the 9th and 10th data points in the ordered list.

  4. Interpolate between the 9th and 10th data points:

    • The 9th data point is x9=60.2x_9 = 60.2,
    • The 10th data point is x10=69.9x_{10} = 69.9.

    To interpolate, we use the formula: Percentile value=x9+(x10x9)×(9.359)\text{Percentile value} = x_9 + (x_{10} - x_9) \times (9.35 - 9) Percentile value=60.2+(69.960.2)×0.35\text{Percentile value} = 60.2 + (69.9 - 60.2) \times 0.35 Percentile value=60.2+9.7×0.35=60.2+3.395=63.595\text{Percentile value} = 60.2 + 9.7 \times 0.35 = 60.2 + 3.395 = 63.595

So, the 85th percentile of the data is approximately 63.6.

Would you like more details on the interpolation method or percentile calculations?

Here are five questions to expand on this topic:

  1. How are percentiles different from quartiles?
  2. How does interpolation work in percentile calculations?
  3. What are some applications of percentiles in real-world data analysis?
  4. How would the method change if the data weren't already sorted?
  5. How does the choice of formula affect percentile calculation methods?

Tip: When calculating percentiles, always make sure the data is sorted in ascending order before applying the percentile formula.

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Math Problem Analysis

Mathematical Concepts

Percentiles
Statistics
Interpolation

Formulas

Percentile rank: Rank = P * (N + 1)
Interpolation formula: Percentile value = x_low + (x_high - x_low) * (Rank - floor(Rank))

Theorems

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Suitable Grade Level

Grades 10-12