Maths Exercises
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Maths Exercices
 
Statistics

Central Tendency Measures
Measures of Dispersion
Measures of skewness and kurtosis

Quartiles

The values that divide the data set into 4 equal parts after it has been arranged in ascending orden are called quartiles.

Split into quartes the following set of data:
2 1 3 2 1 1 2 3 1 4 3 1

Step 1. Rearrange the value in ascending order:
1 1 1 1 1 2 2 2 3 3 3

Step 2. Split into quartes:


This are the same data as before, but this tim eit's split into quartes:

The quartiles are:

  1. Q1 , the lowest quartile is known as the lower quartile, or first quartile. The lowest 25% of the data being found below the first quartile value. It is the same as P25.
  2. Q2 , the quartile in the middle is known as the median. The lowest 50% of the data set should be found below the second quartile. It is the same as the median, D5 and P50.
  3. Q3 , the highest quartile is known as the upper quartile. The lowest 75% of the data set should be found below the third quartile. It is the same as P75.

Steps to find quartile values on a data set with n elements:

Step 1. Calculate: and round to the nearest integer. If L falls halfway between two integers, round up. The Lth element is the lower quartile value.

Step 2. Calculate: and round to the nearest integer. If U falls halfway between two integers, round down. The Uth element is the upper quartile value.


Find the quartiles of the data set: {1, 3, 4, 5, 5, 6, 9, 14, 21}

Step 1. n = 9, so which becomes 3 after rounding up. The lower quartile value is the 3rd data point, Q1=4.
Step 2. which becomes 7 after rounding down. The upper quartile value is the 7th data point, Q3 =9. Using this method, the upper and lower quartile values are always two of the data points.
Step 3. Q2=5, because the median is 5.


There is another way to find the quartiles: Turkey's method:

Divide the set into four equal parts (by Turkey's method):
{6, 3, 4, 9, 6, 2, 7, 7, 8, 4, 10}
Step 1. Arrange your data in ascending order:
2, 3, 4, 4, 6, 6, 7, 7, 8, 9, 10


Step 2. Find the median of the data set.
This is Q2=6


Step 3. Find the median of the lower half of the data set (in parenthesis)
(2, 3, 4, 4, 6), 6, 7, 7, 8, 9, 10
This is Q1=4.


Step 4. Find the median of the upper half of the data set (in parenthesis)
2, 3, 4, 4, 6, 6, (7, 7, 8, 9, 10)
This is Q3=8

Quartiles for grouped data.
The formulae for calculating Q1, Q2 and Q3 for grouped data are:

where:

  • is the lower limit of interval (or class) containing
  • is the frequency of the interval containing
  • w is the width of the interval containing
  • cf is the cumulative frequency up to, but not including the interval.


Find Q1 from these grouped data:
Class Limit
Frequency
Cumulative frequency
0-10
2
2
10-20
3
5
20-30
5
10
30-40
2
12
40-50
6
18
50-60
2
20


N/4=5