メタ説明(152): 生スコアをzスコアに変換し、正規曲線確率を読みます。Zスコア計算機は式、zテーブルの検索、シェーディング領域を表示します。
スキーマ: WebApplication + FAQPage + BreadcrumbList
Zスコア計算機は生スコアを平均と標準偏差を用いて標準スコアに変換し、累積確率(左側の面積)、上尾領域、±z以降の二つの尾の領域を報告します。x、μ、σ(またはそれらを推定するためのデータセット)を入力します。計算機はzと正規曲線領域を返します。
標準化は異なるスケールを一つの軸に配置します。72のマークと180 cmの高さは、それぞれの平均からの標準偏差単位で表されることで比較可能です。
生のスコアをZスコアに変換する
zスコアは、その数値が基準分布の平均よりどれだけ標準偏差上または下に位置するかを示します。正のzは平均より上、負のzは平均より下、ゼロは平均の正確な位置を示します。
平均と標準偏差を含む生スコアを入力し、またはデータセットを貼り付けて、計算機がその2つの合計を推定してから選択したxを標準化できるようにします。
生スコア、平均、標準偏差を入力します。代わりにデータセットを貼り付けると、計算機はまずサンプル平均と標準偏差を計算し、その後選択されたxをそれらの推定値に対して標準化できます。
zスコアの公式を適用してください
生スコアから平均を引いてから標準偏差で割ると、zはxマイナスμに等しく、シグマ全体で計算されます。結果は単位なしで、標準偏差の数として読み取られます。
シグマがゼロの場合、すべての値は平均に等しく、標準化は定義されていません。計算機はその場合をゼロで割らず、遮断します。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.