Running the analysis, and chapter reference

Keywords

JASP normality check, PocketStat, z-scores in JASP, Shapiro-Wilk in JASP, reporting normality, statistics glossary, formula review

Parts 1 to 3 worked every z-score and every share by hand, one value at a time. The software does the same arithmetic for all 200 customers at once.

That leaves you the judgements: whether the shape supports the normal model, which values deserve a second look, and what the result means for the branch. This page runs the chapter’s analysis in JASP and in PocketStat, then turns the output into sentences a manager can use.

The key terms and the formula review at the foot cover all three parts, with every term linked back to the section that defines it.

Running the analysis

Each route reaches the same answers. The two tools differ in two small ways, noted as they arise: PocketStat has no Q-Q plot, and the two tools calculate skewness with slightly different formulas. Use whichever suits the machine you have, and name it in your write-up.

In JASP

Set up the output once: Open Preferences from the menu at the top left and choose Results. Tick Fix the number of decimals and set it to 3, so every table in your output carries the same precision. Tick Display exact p-values, so every p-value is printed in full. JASP prints them in scientific notation: \(4.795 \times 10^{-1}\) is 0.4795, reported as \(p = .480\), and \(1.334 \times 10^{-6}\) is reported as \(p < .001\). Two ways to report a result sets out the reporting style.

Check shape and normality:

Open DMM_Teaching_Data_Full.csv and confirm 200 rows and 26 columns.
Check the measurement icon on every variable you intend to use, as in chapter 1.
Choose Descriptives, then Descriptive Statistics, and move Waiting_Time_Mins, Age, Satisfaction and Spending into Variables.
Under Statistics, tick mean, median, standard deviation, minimum, maximum and quartiles. In the Distribution group, tick Skewness, Kurtosis and Shapiro-Wilk test. JASP adds the standard errors of skewness and kurtosis automatically.
Under Basic plots, tick Distribution plots and its Display density option, then tick Q-Q plots.
Under Customizable plots, tick Boxplots and Label outliers, so the row number of each case beyond the fences appears beside its point.

Create z-scores: Add a computed column, as you did for Wait_Band in chapter 1, name it Wait_z, and use the R formula box:

(Waiting_Time_Mins - mean(Waiting_Time_Mins)) / sd(Waiting_Time_Mins)

Wait_z holds each customer’s z-score. Leave its type as Scale: a z-score is a measurement with equal gaps, and you will want its mean, minimum and maximum. Run Descriptive Statistics on it once as a check. The mean should be 0 and the standard deviation 1. To find the cases that stand out, tick Boxplots and Label outliers under Customizable plots for Wait_z. The axis reads in z-scores, and each point beyond the whiskers carries its row number.

Annotate and export: Click the small arrow beside any analysis title in the output and choose Add Note. Write one or two sentences saying what the table or plot shows and what follows for the manager. To save the whole output with its notes, choose Export Results from the file menu.

In PocketStat

PocketStat runs inside this page. Nothing installs, nothing uploads, and your data stays on your own machine.

If the panel above stays blank, open PocketStat in a new tab instead. The tool needs to be served over the web, so it stays blank when this page is opened as a file from disk.

Load DMM_Teaching_Data_Full.csv from the Data tab and check the type shown for each variable.
Choose Distribution shape from the technique list and pick Waiting_Time_Mins. The results give skewness, its standard error, the skewness ratio labelled z (skewness), and the Shapiro-Wilk \(W\) and p-value. The plot is a histogram with a fitted normal curve.
Choose Z-scores & empirical rule and pick the same variable. The results give the mean, the standard deviation, the percentage of cases within one, two and three standard deviations beside the empirical rule’s 68, 95 and 99.7, and the number of cases beyond 2. The note beneath names the row with the largest z-score. The plot shows the histogram with the standard deviation bands marked, beside a boxplot.
Choose Descriptive statistics for the median and the interquartile range.

PocketStat has no Q-Q plot and gives no z-score for each row. To find the z-score of a particular customer, read the mean and standard deviation from the results and apply \(z = (x - \bar{x}) / s\) yourself. Its skewness differs from JASP’s in the third decimal place, for the reason given in part 1.

Reporting the result

A normality check and a z-score are finished when each becomes a sentence a manager can act on. A written report also needs the statistics themselves, in the standard form set out in Two ways to report a result.

For a normality check: Name the variable, the evidence and the consequence.

Waiting time is close to normal: the histogram has a single symmetric peak, the mean and median differ by half a minute, and a formal test finds no departure from a normal shape. The normal model can be used to estimate shares of customers.

Spending is right-skewed, so typical spending is reported as the median, 494 USD, and cut-offs are taken from the data percentiles.

For a single value: State the value, how far out it sits in plain words, and what to do.

One customer waited 90 minutes, nearly twice the average and 3.7 standard deviations above it. Waits that far out should occur fewer than 3 times in 1,000 customers, so the record is being checked before the figure is used.

For a share from the normal model: State the share, the model it came from, and how well the model matched the data.

About 11 per cent of customers wait 60 minutes or longer under a normal model fitted to this month’s waits. The data hold 21 such customers, 10.5 per cent, so the model and the records agree.

For the whole question: Bring the pieces together and name the limit.

Waiting times at this branch were close to normal over the month, with a mean of 45.5 minutes and a standard deviation of 11.9. Nine customers in ten were seen within about an hour: 90.5 per cent in the data, and 88.9 per cent under the normal model. A one-hour promise would therefore hold, with a margin of one or two customers in 200. One wait of 90 minutes is extreme and is being checked. These figures describe one branch in one month.

Two ways to report a result

Every result in this module is written for two readers. A manager reads for the decision and skips the statistics. A report reader, such as a tutor, an auditor or an analyst checking your work, needs the statistics themselves, set out in a standard form so they can be checked. This module uses APA style, the format set by the American Psychological Association in its 7th edition and used across business and social science reporting.

The same result, written for each reader:

Table 2.18: One result, two readers

Result	For the manager	For the report, in APA style
Waiting time, normality	Waiting times are close to normal, so the normal model can estimate how many customers wait over an hour.	A Shapiro-Wilk test indicated that waiting times were consistent with a normal distribution, \(W = .993\), \(p = .480\).
Spending, shape	Spending is right-skewed, so the typical spend is the median, 494 USD.	Spending was positively skewed, skewness \(= 0.64\), \(SE = 0.17\), and departed from normality, \(W = .948\), \(p < .001\).
Waiting time, centre and spread	The average wait was about 45 minutes, and most waits fell within 12 minutes of that.	Waiting times had a mean of 45.46 minutes (\(SD = 11.91\)).
One unusual wait	One customer waited 90 minutes, far longer than anyone else, and that record is being checked.	The longest wait, 90 minutes, lay 3.74 standard deviations above the mean (\(z = 3.74\)).

Six rules cover every APA sentence in this chapter:

Name the test and the variable in words, then give the statistics at the end of the sentence, after a comma or inside brackets.
Set statistical symbols in italics: \(M\), \(SD\), \(SE\), \(W\), \(p\), \(z\).
Drop the zero before the decimal point for a statistic that can never exceed 1. Write \(p = .480\) and \(W = .993\). Keep the zero for a statistic that can exceed 1, such as a mean, a standard deviation, a skewness or a z-score: skewness \(= 0.64\).
Report p to three decimal places: \(p = .480\), \(p = .008\). Below 0.001, write \(p < .001\), because a p-value of exactly zero is impossible.
Report most other statistics to two decimal places: \(M = 45.46\), \(SD = 11.91\), \(z = 3.74\). Test statistics that can only run from 0 to 1, such as \(W\), take three.
Keep the units in the words, not in the brackets: “a mean wait of 45.46 minutes (\(SD = 11.91\))”. Use the symbol \(M\) when the mean appears only in the brackets.

JASP prints its output with the leading zero, 0.480, and so do the tables and working in this book, because they show the numbers as the software gives them. The APA rules apply to the sentence you write. Chapter 4 adds the reports for t-tests, which follow the same six rules.

Chapter review

Look at the shape before you choose a summary or a model. Skewness and kurtosis put numbers on the shape, and their standard errors tell you whether the numbers exceed sampling variation. The normal distribution is a model fixed by a mean and a standard deviation, and its areas are shares of cases. A z-score places one value inside a distribution in units of standard deviations, and it lets you compare values measured in different units. Check a variable with all five checks before you use the normal model, and for skewed or capped variables report percentiles, the median and the interquartile range taken from the data.

Key terms

Every term links to the section that defines it, and the three groups follow the three parts of the chapter, so this doubles as a revision map.

Part 1, the shape of a distribution

Distribution: the values a variable takes, and how often
Shape: the outline a histogram makes: peaks, symmetry, tails
Unimodal: having one peak
Symmetric: two halves that mirror each other
Ceiling effect: cases piled up at the top of a scale
Skewness: how far a distribution leans to one side
Standard error: how much a statistic varies from sample to sample
Kurtosis: how heavy the tails are, compared with normal
Leptokurtic: heavier tails than normal
Platykurtic: lighter tails and a flatter peak than normal

Part 2, the normal curve and z-scores

Normal distribution: a symmetric bell-shaped model fixed by its mean and standard deviation
Empirical rule: 68, 95 and 99.7 per cent within 1, 2 and 3 SD
Chebyshev’s rule: at least \(1 - 1/k^2\) within \(k\) SD, for any shape
z-score: distance from the mean in standard deviations
Standardising: converting a column to z-scores
Unusual value: a z-score from 2 up to 3, either sign
Extreme value: a z-score of 3 or more, either sign

Part 3, areas, percentiles and normality

Standard normal distribution: the normal distribution with mean 0 and SD 1
Area under the curve: the share of cases between two values
Percentile from the normal model: \(\bar{x} + z s\) for the z that cuts off the share
Q-Q plot: sorted data against normal predictions
Shapiro-Wilk test: a test of how closely data follow a normal pattern
APA style: the standard form for reporting statistics in a written report
p-value: below 0.05, a departure larger than chance

Formula review

Table 2.19: Every formula used in this chapter

Quantity	Formula
z-score, sample	\(z = \dfrac{x - \bar{x}}{s}\)
z-score, population	\(z = \dfrac{x - \mu}{\sigma}\)
Value from a z-score	\(x = \bar{x} + z \times s\)
Skewness, as JASP reports it	\(\dfrac{n}{(n-1)(n-2)} \sum \left(\dfrac{x - \bar{x}}{s}\right)^3\)
Skewness ratio	\(\dfrac{\text{skewness}}{\text{SE of skewness}}\), beyond \(\pm 2\) flags skew
Kurtosis ratio	\(\dfrac{\text{excess kurtosis}}{\text{SE of kurtosis}}\), beyond \(\pm 2\) flags unusual tails
Empirical rule	about 68%, 95%, 99.7% within \(1, 2, 3\) SD
Chebyshev’s rule	at least \(1 - \dfrac{1}{k^2}\) within \(k\) SD, for \(k > 1\)
Share above \(z\)	\(1 -\) area below \(z\)
Share below \(-z\)	area above \(+z\)
Share between \(z_1\) and \(z_2\)	area below \(z_2 -\) area below \(z_1\)
Normal notation	\(X \sim N(\mu, \sigma)\), and \(Z \sim N(0, 1)\)

References and further reading

OpenStax. (2023). Introductory business statistics 2e. Rice University. Chapter 2 covers skewness and chapter 6 covers the normal distribution.
Pallant, J. (2016). SPSS survival manual (6th ed.). Open University Press.
Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3-4), 591-611.
Villa College. (2026). BUSS2111 Decision Making for Management module handbook 2026/27.