To find Q1Q1Q1 and Q3Q3Q3, you first sort your data, then use either a position formula or the “median of halves” idea to locate the 25% and 75% points in the ordered list.

What Q1 and Q3 Mean

  • Q1 (first quartile) : A value where about 25% of the data are at or below it, and 75% are above.
  • Q3 (third quartile) : A value where about 75% of the data are at or below it, and 25% are above.
  • Together with the median Q2Q2Q2, they split your data into four equal parts.

Method 1: Median-of-Halves (Most Common in Classes)

This is the simple, “by hand” method many school courses use.

Step-by-step

  1. Sort the data
    Arrange all values from smallest to largest.

  2. Find the median (Q2)

    • If there are an odd number of data points, the median is the middle value.
    • If there are an even number, the median is the average of the two middle values.

  3. Split into lower and upper halves

    • For an odd number of data points:
      • Exclude the median itself.
      • Lower half = all data below the median.
      • Upper half = all data above the median.
    • For an even number of data points:
      • Lower half = the first n/2n/2n/2 values.
      • Upper half = the last n/2n/2n/2 values.

  4. Find Q1 and Q3

    • Q1Q1Q1 = median of the lower half.
    • Q3Q3Q3 = median of the upper half.

Quick example (odd number of values)

Data: 3,5,7,8,103,5,7,8,103,5,7,8,10

  1. Sorted already.
  2. Median Q2=7Q2=7Q2=7 (middle value).
  3. Lower half: 3,53,53,5; upper half: 8,108,108,10.
  4. Q1=Q1=Q1= median of 3,5=(3+5)/2=43,5=(3+5)/2=43,5=(3+5)/2=4.
    Q3=Q3=Q3= median of 8,10=(8+10)/2=98,10=(8+10)/2=98,10=(8+10)/2=9.

Method 2: Position Formulas (More “textbook” / software-like)

Another common approach is to compute the position of each quartile in the ordered list and then read (or interpolate) the value there. You’ll often see formulas like:

  • Position of Q1Q1Q1: n+14\dfrac{n+1}{4}4n+1​th value
  • Position of Q2Q2Q2 (median): n+12\dfrac{n+1}{2}2n+1​th value
  • Position of Q3Q3Q3: 3(n+1)4\dfrac{3(n+1)}{4}43(n+1)​th value

Here nnn is the number of data points (after sorting).

How to use these positions

  1. Sort the data.

  2. Compute the position :

    • Example: n=11n=11n=11.
      • Q1Q1Q1 position = (11+1)/4=3(11+1)/4=3(11+1)/4=3 → 3rd value.
      • Q3Q3Q3 position = 3(11+1)/4=93(11+1)/4=93(11+1)/4=9 → 9th value.

  3. If the position is an integer :
    Take the value at that position.

  4. If the position is not an integer (e.g. 7.5):

    • Let the integer part be kkk.
    • QQQ lies between the kkkth and (k+1)(k+1)(k+1)th data values.
    • Interpolate:

Q=low+fraction×(high−low)Q=\text{low}+\text{fraction}\times (\text{high}-\text{low})Q=low+fraction×(high−low)

where:

   * low=\text{low}=low= value at position kkk,
   * high=\text{high}=high= value at position k+1k+1k+1,
   * fraction=\text{fraction}=fraction= decimal part (e.g. 0.5 from 7.5).

This is similar to how some universities and software define quartiles.

Odd vs Even Number of Data Points

If nnn is odd

  • Median is a single value in the middle.
  • Median-of-halves method:
    • Exclude that value when forming lower/upper halves.
  • Position method:
    • Positions will naturally tell you if Q1 or Q3 falls exactly on a data point or between two.

If nnn is even

  • Median is the average of the two middle values.
  • Median-of-halves method:
    • Lower half = first n/2n/2n/2 values, upper half = last n/2n/2n/2 values.
  • Position method:
    • Same formulas, but positions are more likely to be non-integers, so interpolation is used.

Example Walkthrough (Even Number of Values)

Data: 4,6,8,11,13,18,21,254,6,8,11,13,18,21,254,6,8,11,13,18,21,25 (already sorted, n=8n=8n=8)

  1. Median (Q2): average of 4th and 5th values
    (11+13)/2=12(11+13)/2=12(11+13)/2=12.

  2. Lower half: 4,6,8,114,6,8,114,6,8,11

    • Median of lower half: average of 6 and 8
      Q1=(6+8)/2=7Q1=(6+8)/2=7Q1=(6+8)/2=7.

  3. Upper half: 13,18,21,2513,18,21,2513,18,21,25

    • Median of upper half: average of 18 and 21
      Q3=(18+21)/2=19.5Q3=(18+21)/2=19.5Q3=(18+21)/2=19.5.

So Q1=7Q1=7Q1=7, Q2=12Q2=12Q2=12, Q3=19.5Q3=19.5Q3=19.5.

Different “Correct” Answers in Different Sources

You might notice that textbooks, teachers, and calculators sometimes give slightly different values for Q1 and Q3 for the same data. That’s not you being wrong; there are multiple standard definitions used in statistics.

  • Some use the median-of-halves method.
  • Others use position formulas with interpolation.
  • Statistical software (like R, Excel, Python) can use one of several built-in algorithms.

If you’re working on homework or an exam:

  • Check your textbook’s method or your teacher’s instructions.
  • Stick to that method consistently.

Extra: Quick “Recipe” You Can Memorize

If you want a compact checklist:

  1. Sort data.
  2. Find median Q2Q2Q2.
  3. Form lower and upper halves.
  4. Take median of lower half → Q1Q1Q1.
  5. Take median of upper half → Q3Q3Q3.

If you’d like, send me a specific data set and I can walk you through finding Q1Q1Q1 and Q3Q3Q3 step by step for that exact list.