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Statistics basics: z-scores, percentiles, quartiles and combinations

Three statistics skills that come up again and again: turning a z-score into a percentile, finding quartiles in a data set, and counting with the combination formula.

2 دقیقه مطالعهبه‌روزرسانی ۱۱ آبان ۱۴۰۵نوشته Sumzo teamبازبینی: Sumzo team

A z-score says how many standard deviations a value is above or below the mean. To find a percentile from a z score, look up the area to the left of z under the standard normal curve: that area, as a percentage, is the percentile. This lesson shows how, then covers two more everyday tools — quartiles and combinations.

Step 1: find the z-score

z=x−μσz = \frac{x - \mu}{\sigma}
$x$ is the value, $\mu$ the mean and $\sigma$ the standard deviation.

مثال حل‌شده

Easy: a test score

A test has mean 70 and standard deviation 10. Find the z-score of a mark of 85.

نمایش راه‌حل
  1. Subtract the mean and divide by the standard deviation.
    z=85−7010=1.5z = \frac{85 - 70}{10} = 1.5

پاسخ: z=1.5z = 1.5: the mark is 1.5 standard deviations above the mean.

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Step 2: how to find percentile from z score

For data that is approximately normally distributed, a standard normal table gives the proportion of values below each z-score. Some standard values: z=−1z = -1 → 15.87%, z=−0.5z = -0.5 → 30.85%, z=0z = 0 → 50%, z=1z = 1 → 84.13%, z=1.5z = 1.5 → 93.32%, z=1.96z = 1.96 → 97.50%, z=2z = 2 → 97.72%.

  1. Find the z-score.
  2. Look up the area to the left of zz in a standard normal table (or use a statistics calculator).
  3. Multiply by 100: that is the percentile. For the mark of 85, z=1.5z = 1.5 gives about the 93rd percentile.
  4. For a negative zz, the table gives a small area: z=−0.5z = -0.5 is about the 31st percentile.

How to find quartiles in a data set

مثال حل‌شده

Medium: quartiles of nine values

Find the quartiles of 3, 7, 8, 5, 12, 14, 21, 13, 18.

نمایش راه‌حل
  1. Sort the data.
    3, 5, 7, 8, 12, 13, 14, 18, 213,\ 5,\ 7,\ 8,\ 12,\ 13,\ 14,\ 18,\ 21
  2. The median (Q2) is the middle value: 12.
  3. Q1 is the median of the lower half (3, 5, 7, 8): 5+72=6\frac{5 + 7}{2} = 6. Q3 is the median of the upper half (13, 14, 18, 21): 14+182=16\frac{14 + 18}{2} = 16.
  4. The interquartile range is 16−6=1016 - 6 = 10.

پاسخ: Q1 = 6, median = 12, Q3 = 16, IQR = 10

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The combination formula

(nr)=n!r! (n−r)!\binom{n}{r} = \frac{n!}{r!\,(n - r)!}
The number of ways to choose $r$ items from $n$ when order doesn’t matter.

مثال حل‌شده

Hard: choosing a team

In how many ways can you choose 3 people from 10? And how many ordered line-ups of 3 are there?

نمایش راه‌حل
  1. Ordered (permutations): 10×9×8=72010 \times 9 \times 8 = 720.
  2. Each group of 3 can be ordered in 3!=63! = 6 ways, so divide.
    (103)=7206=120\binom{10}{3} = \frac{720}{6} = 120

پاسخ: 120 groups (720 ordered line-ups)

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Common mistakes

  • Dividing by the variance instead of the standard deviation when finding zz.
  • Reading the area to the right of zz as the percentile.
  • Finding quartiles without sorting the data first.
  • Using combinations when order matters (a race finish) — that needs permutations.

پرسش‌های متداول

How do you find a percentile from a z score?

Look up the area to the left of the z-score in a standard normal table and multiply by 100. For example, z=1z = 1 gives 0.8413, the 84th percentile.

How do you find quartiles in a data set?

Sort the data, find the median, then find the median of each half: the lower half gives Q1 and the upper half gives Q3. (Different textbooks treat the middle value slightly differently.)

When do I use the combination formula?

When you choose items and the order doesn’t matter — a committee, a hand of cards, lottery numbers.

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