Z Table, Class Width and Other Statistics Basics (With Examples)

Interactive z table: type a z-score, get the probability

A z table statistics lookup answers one question: given a z-score, what share of a standard normal distribution falls on either side of it? This z chart for statistics does the lookup instantly, both directions, so you don't have to hunt across a printed z table stats grid for the right row and column.

Left tail, right tail or between two z-scores

A left-tail lookup returns the cumulative probability below your z-score. A right-tail lookup subtracts that from 1. A between lookup subtracts the smaller result from the larger one — same table, different arithmetic.

The curve, shaded to match your lookup

Every result redraws the standard normal curve and shades the exact area under the curve for your query — a left-tail query shades from the far left to your z-score; a between query shades only the middle strip. Computing an area this way is the same skill as an area problem on geometry help, just under a curve instead of a polygon.

Reverse lookup: probability in, z-score out

Type a probability instead of a z-score to run the lookup backward — the case you need for confidence intervals and critical values. The middle 90% of a standard normal distribution sits between $z=-1.645$ and $z=1.645$. The middle 95% sits between $z=-1.96$ and $z=1.96$. The middle 99% sits between $z=-2.576$ and $z=2.576$.

The full standard normal table (positive and negative z)

The standard normal table below covers $z=0.0$ through $z=3.0$ in 0.1 steps, negative and positive, as real text you can search — the same lookup MyStatLab and MyLab Statistics assignments ask for under a different name. GradeDraft is independent of and not affiliated with Pearson or MyStatLab.

Negative z table (z from -3.0 to 0)

Positive z table (z from 0 to 3.0)

How to read a z chart: three worked lookups

Reading a z statistic chart is arithmetic once you know which operation each phrasing calls for. The standard normal distribution — sometimes called the Gaussian distribution, after Carl Friedrich Gauss — is always centered at $z=0$ with total area under the curve equal to 1.

P(Z < 1.96)

This is a left-tail lookup. Find the row nearest $z=1.9$ and interpolate the hundredths place: $P(Z<1.96) = 0.9750$. Roughly 97.5% of a standard normal distribution lies below $z=1.96$.

P(Z > -0.85)

A right-tail lookup starts from the complement rule: $P(Z>z) = 1 - P(Z<z)$. First find $P(Z<-0.85) = 0.1977$, then subtract from 1: $P(Z>-0.85) = 1-0.1977 = 0.8023$.

P(-1.2 < Z < 2.1)

A between lookup subtracts the smaller cumulative probability from the larger one: $P(-1.2<Z<2.1) = P(Z<2.1) - P(Z<-1.2) = 0.9821 - 0.1151 = 0.8670$. About 86.7% of the distribution sits between those two z-scores.

Turning a raw score into z first: (x - mu) / sigma

Raw scores need converting before any lookup applies. The z-score formula is $z = \dfrac{x-\mu}{\sigma}$: subtract the mean, then divide by the standard deviation. A test score of 82, with $\mu=74$ and $\sigma=8$, converts to $z=\dfrac{82-74}{8}=1.0$ — read off the positive table as $P(Z<1.0)=0.8413$. Solving that formula for $x$ instead of $z$ is the identical algebra in solving equations step by step.

How do you find the class width in statistics?

The formula: (max - min) / number of classes, rounded up

Class width statistics questions all use one formula: $\text{class width} = \dfrac{\text{max}-\text{min}}{\text{number of classes}}$, rounded up to a convenient value. Sturges' rule, $1+3.322\log_{10}n$, is one common way to pick the number of classes before you compute the width, if the assignment doesn't specify one.

A worked example from a raw data set

Twelve exam scores: 62, 65, 70, 71, 74, 78, 81, 85, 88, 91, 95, 99. Range: $99-62=37$. With 5 classes, the raw width is $37 \div 5 = 7.4$, rounded up to 8.

Class width with decimals: why you round up, never down

Rounding down would leave the highest data point, 99, outside every class. How do you find the class width in statistics when the division doesn't come out even? Always round up: a class width of 7 (rounded down from 7.4) only reaches 62+7×5=97, one class boundary short of covering 99.

Reading class width off a histogram

When the class width is not uniform

Not every frequency table uses one width throughout. Household income brackets are a common example: $0–$24,999 and $25,000–$49,999 (width 25,000 each), then $50,000–$99,999 (width 50,000), then an open-ended $100,000+ class. Unequal widths group the data more evenly when most values cluster at one end and thin out at the other; class width is not uniform whenever a data set has that kind of skew.

What is a population in statistics - population vs sample

Population: every member of the group in question

A population is every member of the group a question is actually about — every enrolled student, every part off the line, every voter. What is a population in statistics comes down to that one word: every.

Sample: the part you actually measured

A sample is the subset you actually collected data from. Surveying 400 of a university's 22,000 students is a sample; the 22,000 students are the population.

Population characteristics and why they are usually unknown

A population characteristic — the true mean, the true proportion, the true standard deviation — is almost never fully knowable, because measuring an entire population is rarely practical. That's the entire reason sampling exists: a sample statistic estimates a population characteristic you can't measure directly. A historical census schedule is a rare case where the whole population actually was counted; see history homework help for how to source and cite one.

Which of the following best describes a population (how these exam items are written)

Exam questions phrase this as a multiple-choice pair: one option names a specific measured group (a sample), the other names the entire group of interest (the population). The tell is scope — "surveyed," "measured," or a specific count signals a sample; "all," "every," or "entire" signals a population.

A statistic describes the sample; a parameter describes the population

The statistic is a number describing the sample. A parameter is the matching number that describes the population. The two words track the same distinction as population vs sample, applied to the summary numbers instead of the groups themselves.

The mu / x-bar and sigma / s notation pairs

Notation keeps the two straight without restating "population" or "sample" every time: $\mu$ (population mean) pairs with $\bar{x}$ (sample mean); $\sigma$ (population standard deviation) pairs with $s$ (sample standard deviation); $N$ (population size) pairs with $n$ (sample size); $p$ (population proportion) pairs with $\hat{p}$ (sample proportion).

Parameter vs statistic in one table

What makes something unusual in statistics?

The z-score rule of thumb: |z| > 2

What makes something unusual in statistics has a standard cutoff: a z-score with $|z|>2$. Under the empirical rule, about 95% of values in a roughly normal distribution fall within two standard deviations of the mean, so anything past that boundary is unusual by definition, not by judgment call.

The range rule of thumb

The range rule of thumb estimates a standard deviation when you don't have one: $s \approx \dfrac{\text{range}}{4}$. Combined with the z-score rule, the "usual" range for individual values runs from roughly $\text{mean}-2s$ to $\text{mean}+2s$ — the same logic behind stating a measurement's uncertainty in physics homework help.

Unusual vs significant vs outlier - three different words

These three get used interchangeably, but they answer different questions. Unusual describes one value relative to its own distribution ($|z|>2$). Statistically significant describes a result unlikely to occur by chance under a null hypothesis, usually judged against $p<0.05$. An outlier is a value far from the rest of a specific data set, often flagged with the 1.5×IQR rule — below $Q_1-1.5\,\text{IQR}$ or above $Q_3+1.5\,\text{IQR}$.

Discrete probability distributions: the other lookup students arrive for

What a discrete probability distribution has to satisfy

A discrete probability distribution has to meet two conditions: every probability is between 0 and 1, and all the probabilities add up to exactly 1.

Finding the missing probability so the column sums to 1

The four probabilities must sum to 1: $0.1+0.3+0.4=0.8$, so the missing value is $1-0.8=0.2$.

Calculating the mean of a discrete probability distribution

The mean of a probability distribution is the sum of the last column: $\mu = \sum x \cdot P(x) = 0+0.30+0.80+0.60 = 1.7$. This is the same expected-value calculation behind pricing an insurance product or a business decision tree; see accounting, finance and economics homework for that application.

Quick-reference formula sheet

When the stats assignment is due and the table is only step one

A z-score, a class width, or a missing-probability question takes minutes with the table above. A full problem set — hypothesis tests, confidence intervals, a dataset in SPSS or R — is where most students look for homework help instead of working every step alone, and GradeDraft's statistics desk takes those on directly.