Z-점수 계산기는 평균과 표준편차를 사용하여 원시 점수를 표준 점수로 변환한 후, 누적 확률(왼쪽 영역), 상단 꼬리 영역, ±z 이후의 두 꼬리 영역을 보고합니다. x, μ, σ(또는 이를 추정하는 데이터 세트)를 입력합니다; 계산기는 z와 정상 곡선 영역을 반환합니다.
표준화는 서로 다른 척도를 하나의 축에 배치합니다. 72 표시와 180 cm 높이는 각각 자신의 평균에서 표준편차를 가진 단위로 표현될 때 유사해집니다.
원시 점수를 z-점수로 변환하세요
z-점수는 어떤 값이 기준 분포의 평균보다 몇 개의 표준편차에 위치하는지를 나타냅니다. 양의 z는 평균 위를 의미하고; 음의 z는 그 아래를 의미하며; 0은 정확히 평균에 위치함을 의미합니다.
평균과 표준편차가 포함된 원시 점수를 입력하거나, 데이터 세트를 붙여넣어 계산기가 두 합계를 추정할 수 있도록 하여 선택한 x를 표준화하세요.
원시 점수, 평균, 표준편차를 입력합니다. 대신 데이터 세트를 붙여넣으면, 계산기는 먼저 표본 평균과 표본 표준편차를 계산한 뒤, 선택한 x를 그 추정치에 대해 표준화할 수 있습니다.
z-점수 공식을 적용하세요
원평균에서 평균을 빼고 표준편차로 나누면 z는 x-mu이며, 시그마 전체에 해당합니다. 결과는 단위가 없으며, 표준편차의 수로 읽힙니다.
시그마가 0이면 모든 값은 평균과 같고 표준화가 정의되지 않습니다; 계산기는 0으로 나누는 대신 그 경우를 차단합니다. z = (x − μ) / σ ```
Subtract the mean from the score, then divide by the standard deviation. The result is unitless: "1.5 standard deviations above the mean," not "1.5 marks."
If σ = 0, every value equals the mean and standardisation is undefined. The calculator blocks that case rather than dividing by zero. Use population μ and σ when they are known; use sample estimates when only sample data exist, and say so in any report.
Convert a z-score to a probability
Under a standard normal curve, each z maps to a cumulative probability equal to the area to the left of that z. The calculator also reports the area to the right and the two-tail area beyond plus or minus z.
A z of 1.00 has left area about 0.8413, so roughly 84 percent of a normal population lies at or below that standardised score.
- Area to the left (percentile-style cumulative probability)
- Area to the right = 1 − left area
- Two-tail area = 2 × smaller one-tail area (for |z|)
A z of 1.00 has left area ≈ 0.8413, so about 84.13% of a normal population lies at or below that score. Right area ≈ 0.1587.
This page stops at the area. Deciding whether a result is "significant" at α = 0.05 is the P-Value Calculator's job.
Find the probability between two z-scores
For two ordered z values, the area between them equals the cumulative area at the larger z minus the cumulative area at the smaller z. Between minus 1 and plus 1 that difference is about 0.6826, matching the empirical rule claim that roughly 68 percent of normal data sits within one standard deviation of the mean.
Example: between z = −1 and z = 1, area ≈ 0.8413 − 0.1587 = 0.6826, matching the empirical rule's "about 68% within one standard deviation." Between 0 and 1.96 the area is about 0.4750 each side of the mean for a central 95% interval construction.
Calculate the z-score for a raw score of 5
Given mean 3 and standard deviation 2, the standardised score is five minus three, divided by two, which equals one. Left area is about 0.8413 and right area about 0.1587. On a dual-axis normal curve the raw axis marks 3 at the centre and 5 at the plus-one sigma line, so the same shade answers both questions.
z = (5 − 3) / 2 = 2 / 2 = 1z = 1. Left area ≈ 0.8413. Right area ≈ 0.1587. The score sits one standard deviation above the mean. On a dual-axis normal curve, the raw axis marks 3 at the centre and 5 at the +1σ line.
Read the z-table
Printed z-tables usually list cumulative area from the far left up to z, with rows for the first decimal and columns for the second (some tables show area from the mean out to z instead). Look up z = 1.00 on a cumulative table: row 1.0, column .00 → 0.8413.
Negative z values use symmetry. The left-tail area at minus a equals one minus the left-tail area at plus a. Many printed tables only show positive z for that reason. Looking up 1.96 and subtracting from one is the standard route to the lower-tail area at minus 1.96, about 0.0250.
Two conventions cause most student errors. Confirm whether your table is cumulative from −∞ or area from 0 to z. The calculator's left-area output matches the cumulative-from-−∞ convention. For negative z, use symmetry: P(Z ≤ −a) = 1 − P(Z ≤ a), or read the positive entry and subtract from 1.
Read the normal curve
The standard normal curve is bell-shaped, centred at 0, with inflection points near ±1. Shaded regions correspond to the probabilities above. A dual axis can show raw units underneath (μ at 0, μ+σ at +1) so the same shade answers both "z > 1" and "x > 5" in the worked example.
Shaded two-tail regions for a critical z of 1.96 each hold about 2.5 percent of the area, totaling 5 percent. That picture bridges into confidence intervals and p-values, but those decision tools still live on their own pages so this one stays focused on standardisation and area.
The curve assumes normality. Strong skew or heavy outliers make the area translation unreliable. Check a histogram or stem-and-leaf plot before treating the probability as exact for real data.
Interpret a z-score
Under approximate normality, a z near zero is typical, near one is common, near two is unusual, and near three is rare under the empirical rule. Those labels are descriptive distances, not automatic hypothesis-test decisions. Whether a result rejects a null at a chosen alpha depends on tails and study design on the P-Value Calculator.
| z | Reading | ||
|---|---|---|---|
| ≈ 0 | Typical, near the mean | ||
| ≈ 1 | About one SD away; common | ||
| ≈ 2 | Unusual; about 5% combined in both tails beyond ±2 | ||
| ≈ 3 | Rare; about 0.3% beyond ±3 under the empirical rule |
"Unusual" is descriptive here, not a hypothesis-test decision. A z of −2.3 means the value sits 2.3 standard deviations below the mean; whether that rejects a null hypothesis depends on α, tails and study design on the p-value page.
Standardisation without over-claiming
Computing z does not require normality. Translating z into a table probability does. Keep those steps separate when a histogram shows strong skew or when a single outlier dominates the sample standard deviation. In those cases report the z as a descriptive distance and be cautious about the area.
When only sample estimates of the mean and standard deviation are available, say so. For small samples and unknown sigma, t-based inference is often the better follow-on tool; the Confidence Interval Calculator defaults to t in that setting.
Frequently asked questions
What is a z-score?
A z-score (standard score) states how many standard deviations a raw value sits above or below the mean: z = (x − μ) / σ. Positive means above; negative means below.
Can a z-score be greater than 3?
Yes. Large |z| values are uncommon under a normal model but appear with outliers, heavy tails or wrong σ. They are valid outputs; they are also a prompt to check the data and the normality assumption.
How do I convert z back to a raw score?
x = μ + zσ. For μ = 3, σ = 2, z = 1: x = 3 + 2 = 5.
What probability does the calculator show?
By default, the cumulative area to the left of z under the standard normal curve, plus the right-tail and two-tail areas derived from it.
Is a z-score the same as a p-value?
No. A z-score is a standardised distance. A p-value is the tail probability used in a significance decision. This page computes areas; the P-Value Calculator handles α comparisons.
Which z-table convention should I use?
Prefer cumulative-from-−∞ tables, which match most software and this calculator's left area. If your table shows area from the mean to z, add 0.5 for positive z to get the cumulative left area.
What if I only have sample data?
Use the sample mean and sample standard deviation as estimates, and state that z is approximate. For small samples and unknown σ, t-based methods may fit better for inference (see the Confidence Interval Calculator).
Does standardisation assume a normal distribution?
Computing z = (x − μ) / σ does not. Translating z into normal-table probabilities does. Keep those steps separate when the data are clearly non-normal.
Summary
The Z-Score Calculator converts a raw score to z = (x − μ) / σ and reports the matching standard-normal areas: left, right and two-tail. The worked case x = 5, μ = 3, σ = 2 yields z = 1 and left area ≈ 0.8413.
Use the z-table or the shaded curve to read probabilities; use the P-Value Calculator when the question is whether to reject a null at a chosen α. Standardisation enables comparison across scales; probability translation still needs an approximate normal model.