Math Problem Statement

Show working out for I.Q.R

Show working out for I.Q.R

Solution

To calculate the IQR (Interquartile Range), let's work step by step based on the data provided in the image.


1. Sort the Data

The given data is already sorted:

1,1,1,1,2,2,2,2,2,3,3,4,5,5,6,7,7,7,7,8,8,8,9,91, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 7, 7, 8, 8, 8, 9, 9


2. Determine the Quartiles (Q1 and Q3)

Step 2.1: Find Q1 (Lower Quartile)

  • Q1Q1 is the median of the lower half of the data (the data before the overall median).
  • Lower half:
    1,1,1,1,2,2,2,2,2,3,3,41, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4
  • Median of this lower half:
    2,2,2,2,2,3,3,42, 2, \mathbf{2}, 2, 2, 3, 3, 4
    The median lies between the 6th and 7th numbers:
    Q1=2+32=2.5Q1 = \frac{2 + 3}{2} = 2.5

Step 2.2: Find Q3 (Upper Quartile)

  • Q3Q3 is the median of the upper half of the data (the data after the overall median).
  • Upper half:
    5,5,6,7,7,7,7,8,8,8,9,95, 5, 6, 7, 7, 7, 7, 8, 8, 8, 9, 9
  • Median of this upper half:
    7,7,7,7,8,8,8,97, 7, \mathbf{7}, 7, 8, 8, 8, 9
    The median lies between the 6th and 7th numbers:
    Q3=7+82=7.5Q3 = \frac{7 + 8}{2} = 7.5

3. Calculate the Interquartile Range (IQR)

The formula for IQRIQR is: IQR=Q3Q1IQR = Q3 - Q1 Substitute the values: IQR=7.52.5=5IQR = 7.5 - 2.5 = 5


Final Answer:

IQR=5\boxed{IQR = 5}

Would you like further clarification or a breakdown of a specific part? 😊


Related Questions:

  1. What is the significance of the interquartile range in statistics?
  2. How is the IQR used to detect outliers in a dataset?
  3. What is the relationship between IQR and the overall range of a dataset?
  4. How can you visually represent Q1, Q3, and IQR using a boxplot?
  5. How does the size of the IQR indicate data dispersion?

Tip:

When calculating medians for quartiles, always double-check if the dataset contains an even or odd number of values in each half! This ensures accurate calculations.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Descriptive Statistics
Quartiles
Interquartile Range (IQR)

Formulas

Q1 (Lower Quartile) = Median of the lower half of the data
Q3 (Upper Quartile) = Median of the upper half of the data
IQR = Q3 - Q1

Theorems

Median Theorem for Ordered Data

Suitable Grade Level

Grades 7-9