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Descriptive and Distribution

Descriptive Statistics

Descriptive statistics summarize your data's typical value, spread and shape.

Descriptive statistics summarize what your data look like. For numeric scores, they describe the typical value, the spread and the shape of the distribution.

Example

How much time do first-year students spend on their phones each day? We asked 120 first-year students to read their phone's weekly screen-time report, and each student's daily average is one score.

One Sample + Continuous Variable → Descriptive Statistics
Estimates
A variable's center, spread and shape
Observations
Independent
Headline Statistic
M
Groups
1

The Basics

When to Use It

Use it first, on every variable, before any test: one sample, one numeric variable, and the question "what do these scores look like?" It gives a typical score, how spread out the scores are, and the shape of their distribution. For example:

  • How much time do first-year students spend on their phones each day?
  • How are PHQ-9 depression scores spread across a primary-care sample, and how many score in the severe range?
  • How fast, and how variable, are people's reaction times on a Stroop task?

This page covers numeric scores. Categories, such as a diagnosis or a yes/no answer, have descriptive statistics too: the count and percentage in each category, and the most common one (the mode). The methods for them are under Categorical Data and Proportions.

Vocabulary You Need

Definition Median (Mdn)

The middle score when the scores are lined up in order, or the average of the middle two. Half the scores fall at or below it and half at or above. Unlike the mean, it barely moves when a few scores are extreme. Glossary →

Definition Standard Deviation (s)

How spread out individual scores are around their mean. For roughly bell-shaped scores, about two-thirds fall within one standard deviation of the mean. Its square, s², is the variance: the average squared distance from the mean (dividing by n − 1). Glossary →

Definition Interquartile Range (IQR)

The 75th percentile (the third quartile) minus the 25th percentile (the first quartile), so the width of the band holding the middle half of the scores. It's the natural partner of the median: neither is moved by how extreme the highest and lowest scores are, so together they describe skewed data better than the mean and standard deviation. It's the box in a box plot. For a normal distribution it's about 1.35 standard deviations. Glossary →

Definition z-Score (z)

A score's distance from the mean, measured in standard deviations: z = (x − x̄) / s. A z of 0 is exactly average, 1 is one standard deviation above, and −2 is two below. Converting to z-scores puts measures with different units on one scale, so a person's anxiety and sleep can be compared. It doesn't change the shape of the distribution: skewed scores stay skewed. In a normal distribution about 95% of z-scores fall between −1.96 and 1.96. Also called a standard score. Glossary →

Definition Skewness

How lopsided a set of scores is. It's 0 for a symmetric distribution, positive when a long tail stretches toward high scores (right skew, as with reaction times or income) and negative when the tail stretches toward low scores (left skew, as with scores on an easy test). It's roughly the average cubed z-score: cubing keeps the sign and lets far-out scores dominate, so a few extreme scores can drive it. With right skew the mean is usually pulled above the median. Programs use different formulas; SPSS and SAS report the sample-adjusted G₁. There's no universal cutoff for “too skewed”. Glossary →

Definition Kurtosis

How heavy a distribution's tails are: how prone it is to rare, extreme scores, compared with a normal curve. It's roughly the average z-score raised to the fourth power, so it's driven almost entirely by the scores far from the mean. Most software reports excess kurtosis, which subtracts 3 so that a normal distribution scores 0; positive values mean the scores are more prone to extremes than a normal curve's, negative values less so. Despite a long textbook tradition, it says almost nothing about whether the peak is pointy or flat. It's even more unstable than skewness in small samples. Glossary →

Choosing and Interpreting Your Summaries

1. Related Scores Not needed to calculate anything; it matters once you read the numbers as describing a population.
What It Means
Descriptive statistics describe whatever scores you have, related or not. Independence matters when you treat the scores as a sample, as a confidence interval for the mean does: several scores from one person, or from people who influence each other, carry less information than the same number of unconnected people.
Applies To
Observations
How to Check
By design, not by a test. Did each person contribute their own data, with no clusters such as roommates, couples, therapy groups or classrooms whose members influence each other? If there are clusters, does the analysis model every one of them?
How Much It Matters
Not at all for the numbers themselves, which describe whatever scores you have. It matters when you read them as describing a population: several scores from one person, or from people who influence each other, make the sample look bigger, and more like many separate people, than it is.

What to Do

  • Design Change Average each person's repeated measurements into one score, as the screen-time study did with a week of daily totals. If how much people vary from day to day is the question, describe that too, as each person's SD across days.
  • Proceed with Caveat If the data are clustered (students within classes), describe the clusters' averages as well as the individuals, and say how many clusters there are.
2. What a Point on the Scale Means The mean and SD read a one-point gap as the same amount anywhere on the scale; the median and percentages don't.
What It Means
The mean, standard deviation, skewness and kurtosis all add and subtract scores, so they treat the gap from 2 to 3 as the same amount as the gap from 8 to 9. Times, counts and totals of many items usually qualify. A single rating item, or ranks, may not: “agree” to “strongly agree” needn't be the same step as “neutral” to “agree”.
Applies To
Outcome
How to Check
By reasoning about the measure, not by a test. Ask whether one more unit means the same amount of the thing everywhere on the scale.
How Much It Matters
Hours of screen time are equal-interval, so no problem here. For a single rating item or ranks, the mean and SD can mislead, and skewness and kurtosis mean little; the median, IQR and each response's percentage don't need equal intervals.

What to Do

  • Robust Estimation Report the median and interquartile range, plus how many people gave each response, which only need the scores to be in order.
  • Design Change Average or sum several items into a scale score, which behaves much more like an equal-interval measure than any single item.
3. The Shape of the Scores With a long tail or two peaks, the mean alone no longer describes a typical score.
What It Means
The mean and standard deviation picture the scores as one balanced hump: most people near the mean, thinning out evenly on both sides. With a long tail on one side the mean is pulled toward the tail and the SD blends a short side with a long one; with two humps the mean can describe almost nobody. The numbers are still correct, just less informative.
Applies To
Outcome
How to Check
Look at a histogram or box plot. Compare the mean with the median: a clear gap means skew. Skewness near 0 supports symmetry. Two humps show up in the histogram, not in skewness or kurtosis: if you see them, look for two groups mixed together, and describe each. Don't use a normality test as a gatekeeper: with small samples it misses real problems, and with large ones it flags trivial ones.
How Much It Matters
Only for what M and SD convey; they're still correct numbers. Screen time is right-skewed (skewness 1.13): the mean sits 0.40 hours above the median. M ± 2 SD runs from 0.50 to 8.50 hours and holds 115 of the 120 students, but all 5 outside it are heavy users (up to 12.2 hours), and its lower end is below every student's score (the minimum is 1.5).

What to Do

  • Robust Estimation Report the median and IQR alongside (or instead of) the mean and SD, and say which way the tail runs.
  • Transformation Log-transforming right-skewed scores such as times often makes them close to symmetric (here, skewness drops to 0.03). Describe the result on the log scale, or convert the mean of the logs back to hours: that's the geometric mean, here 4.11 hours, close to the median.
  • Proceed with Caveat Report the mean and SD for comparison with other studies, add the median and IQR, and show a histogram or box plot.
4. Extreme Scores A few far-out scores move the mean, SD, skewness and kurtosis much more than the median and IQR.
What It Means
A few scores far from the rest can pull the mean, SD, skewness and kurtosis around on their own, so those numbers end up describing the few more than the typical student. An extreme score may be an error or a real, unusual case; only the record can tell which.
Applies To
Outcome
How to Check
Look at a box plot, like the one drawn with the skewness histogram on this page. Its dots are outliers by Tukey's (1977) rule: scores more than 1.5 IQRs beyond the quartiles. Here that's above 5.6 + 1.5 × 2.6 = 9.5 hours, so three students (9.7, 10.4 and 12.2 hours); no one is below the lower fence. Only the 12.2-hour student is more than 3 SDs from the mean. Check each against its record.
How Much It Matters
A lot for the mean, SD, skewness and kurtosis; hardly at all for the median and IQR. Dropping the single 12.2-hour student would move the mean from 4.50 to 4.44 hours, the SD from 2.00 to 1.88, skewness from 1.13 to 0.89 and kurtosis from 1.54 to 0.51, while the median stays at 4.10.

What to Do

  • Proceed with Caveat Check each extreme score against its source. Correct clear errors; keep real scores, and report the key numbers with and without them.
  • Robust Estimation Lead with the median and IQR. To flag outliers, measure each score's distance from the median in median absolute deviations (as R's mad() scales them) rather than in SDs from the mean, which the outliers themselves inflate. Here a cutoff of 3 flags the same three students as the box plot.

Limitations

This Is Not Your Test If…

What It Doesn't Tell You

  • Why. Nothing here explains why some students use their phones three times as much as others; descriptives describe, they don't explain.
  • Whether it's a lot. Whether 4.5 hours a day is too much is a question about sleep, study and well-being, not statistics.
  • Who else. The numbers describe these 120 students. They describe all first-year students only as well as the sample represents them, and screen-time reports count minutes the screen is on, not attention.
  • How stable the shape is. With 120 students, skewness and kurtosis are rough: SPSS gives them standard errors of 0.22 and 0.44, and one student changes kurtosis from 1.54 to 0.51.

Common Mistakes

  • Don'tReport only the mean and SD for clearly skewed scores.

    InsteadAdd the median and IQR, and say which way the long tail runs.

  • Don'tDelete scores just because they're more than 3 SDs from the mean.

    InsteadCheck them against the records, keep real ones, and report results with and without them.

  • Don'tCompare skewness or kurtosis from different programs without checking the formula.

    InsteadSay which version you report. SPSS prints the sample-adjusted G₁ and excess G₂; some R and Python functions default to others.

  • Don'tReport the standard error when readers need to know how much people vary.

    InsteadGive the SD to describe the scores, and the SE or a confidence interval only for how precisely the mean is estimated. Say which one error bars show.

  • Don'tSkip straight to the test without looking at the data.

    InsteadDescribe every variable first, with a histogram, so a typo, a ceiling or a long tail is caught before it drives a result.

  • Don'tName the skew after the side where the scores pile up.

    InsteadName it after the long tail: a tail stretching toward high scores is positive (right) skew.

  • Don'tDescribe high kurtosis as a tall, pointy peak.

    InsteadDescribe it as how prone the scores are to extreme values; high kurtosis usually comes from a few unusually extreme scores (Westfall, 2014).

Result Decoder

How It's Reported

Equation Form

Example

M = 4.50, SD = 2.00, Mdn = 4.10, IQR = 2.60, skewness = 1.13, kurtosis = 1.54

Table 1. Descriptive Statistics for Daily Screen Time (Hours)

Example
nMSDMdnIQRMinMaxSkewnessKurtosis
Daily screen time 1204.502.004.102.601.512.21.131.54

Note. Skewness is the sample-adjusted G₁ and kurtosis the excess G₂ (0 for a normal distribution), as SPSS reports them. Quartiles 3.00 and 5.60 hours.

Full Text (APA Style)

Example

First-year students averaged 4.50 hours of screen time a day (SD = 2.00), but the typical student used a little less: the median was 4.10 hours, and the middle half of students fell between 3.00 and 5.60 hours (IQR = 2.60). Scores ranged from 1.5 to 12.2 hours and were positively skewed (skewness = 1.13, kurtosis = 1.54), with a tail of heavy users; the heaviest, at 12.2 hours, was 3.85 standard deviations above the mean.

  • Give the median and IQR alongside the mean and SD whenever the scores are skewed, and say which way the tail runs.
  • Give n, and the minimum and maximum, so readers can spot impossible values.
  • For several variables, put the descriptives in a table like Table 1, and say which skewness and kurtosis formulas you used.
  • Italicize statistical symbols (n, M, SD, Mdn) and give means and SDs to two decimals, or to whatever precision the measure supports.

Reading Each Piece

The equation form above, piece by piece.

M = 4.50 On average, students spent 4.50 hours a day on their phones.
Mdn = 4.10 Half the students used their phones 4.10 hours a day or less. It's below the mean because a few heavy users pull the mean up.
SD = 2.00 Scores spread about 2 hours around the mean. Squaring makes far-out scores count extra, so the SD is a little larger than the students' average distance from the mean (1.56 hours). About 7 in 10 students sit within one SD of the mean, close to the two-thirds a normal curve would have.
IQR = 2.60 The middle half of students used their phones between 3.00 and 5.60 hours a day, a band 2.60 hours wide.
Skewness = 1.13 The scores lean right, with a long tail of heavy users. That fits with the mean (4.50) sitting above the median (4.10).
Kurtosis = 1.54 The scores are more prone to extremes than a normal curve's, here in the long right tail of heavy users. It says almost nothing about how pointy the peak is (Westfall, 2014).

Calculations

The Running Example

Example

The screen-time study: 120 first-year students, one score each, their average daily screen time over a week, in hours. Nothing is being tested; the goal is to describe the scores.

What Your Data Should Look Like

One row per person and one numeric column. If a person was measured several times, first combine their measurements into one score (the screen-time study averaged seven days), unless the variation within each person is what you want to describe. If you have groups, describe each group separately as well as everyone together.

Example
studentscreen_time_hours
13.2
23.4
33.2
43.6
53.8
62.5
… 114 more rows
nMSD
All1204.502.00

Simulated data.

Download all 120 rows (data.csv)

Napkin Math

The whole test, one idea at a time. Under each heading, a line says why the step matters, and the green note reads the equation in plain English with the screen-time study's numbers. Collapse all to read just the headings: they tell the whole story on their own.

  1. 1. Find the Average Screen Time

    The mean shares the total out equally, as if every student used their phone the same amount. It's the most common single summary, and the one most tests are built on.

    \bar{x} = \dfrac{\sum x_i}{n} =\dfrac{540.0}{120}

    = 4.50 hours

    x_i is one student's hours, and \sum x_i adds up all 120 students' hours, 540.0. n is the 120 students. 540.0 ÷ 120 = 4.50 hours a day. Formulas write the mean as \bar{x} ("x-bar"); reports write it as M.

  2. 2. Find the Student in the Middle

    Line everyone up from lightest to heaviest user and take the middle. Unlike the mean, it doesn't care how far out the heaviest users are. A third average, the mode (the most common score), suits categories better than finely measured scores like these.

    Mdn = \begin{cases} x_{\left(\frac{n+1}{2}\right)} & n \text{ odd} \\[4pt] \dfrac{x_{\left(\frac{n}{2}\right)} + x_{\left(\frac{n}{2}+1\right)}}{2} & n \text{ even} \end{cases} =\dfrac{x_{(60)} + x_{(61)}}{2} = \dfrac{4.1 + 4.1}{2}

    = 4.10 hours

    With n = 120, an even number, no one student is in the middle. (n + 1)/2 = 60.5 falls between the 60th and 61st students in line, x_{(60)} and x_{(61)}, and the median is their average. Both logged 4.1 hours. The median is 4.10 hours, 0.40 below the mean.

  3. 3. Measure How Far Scores Spread from the Average

    To measure spread, look at how far each student is from the mean. Squaring the distances stops the ones above and below from canceling out. Dividing by n - 1 rather than n (Bessel's correction) makes up for measuring the distances from the sample's own mean, which sits closer to its scores than the population mean does. The result is the variance.

    s^2 = \dfrac{\sum (x_i - \bar{x})^2}{n - 1} =\dfrac{475.8}{119}

    = 4.00 squared hours

    Each student's distance from 4.50, squared, adds up to \sum (x_i - \bar{x})^2 = 475.8. Divided by n - 1 = 119, that's a variance of 4.00 squared hours.

  4. 4. Put the Spread Back into Hours

    Squared hours are hard to picture, so take the square root. The standard deviation, named by Pearson (1894), is the spread in hours. It's almost a root-mean-square distance of students from the mean, except that the average of the squared distances divides by n - 1, not n.

    s = \sqrt{\dfrac{\sum (x_i - \bar{x})^2}{n - 1}} =\sqrt{3.998}

    = 2.00 hours

    s (reported as SD) is the square root of that 4.00-squared-hour variance, which is 2.00 hours. Because the distances were squared before they were averaged, far-out scores count extra, so s is a little larger than the students' plain average distance from 4.50, which is 1.56 hours.

  5. 5. Measure the Spread of the Middle Half

    The SD is pulled up by the heaviest users, and the range, highest minus lowest (1.5 to 12.2 hours here), rests on just the two most extreme students. The interquartile range, the box in Tukey's (1977) box plot, ignores both tails and asks how wide the middle half of the students is.

    IQR = Q_3 - Q_1 =5.6 - 3.0

    = 2.60 hours

    Q_1, a quarter of the way up the line, is 3.0 hours, and Q_3, three quarters of the way up, is 5.6 hours. 5.6 − 3.0 = 2.60 hours.

  6. 6. Place the Heaviest User on the Same Scale

    A z-score says how unusual one score is, by counting how many standard deviations it sits from the mean. It's also the building block of skewness and kurtosis.

    z_i = \dfrac{x_i - \bar{x}}{s} =\dfrac{12.2 - 4.50}{2.00}

    = 3.85 SDs

    x_i is the heaviest user's 12.2 hours, 7.70 above \bar{x} = 4.50. Divided by s = 2.00 hours, that's z = 3.85 standard deviations above the mean.

  7. 7. Measure How Lopsided the Scores Are

    Cubing every z-score keeps its sign and lets the far-out scores dominate, so a long tail on one side pushes the total that way. Zero means balanced. Pearson (1895, p. 370) measured skewness differently, as the distance between the curve's peak and its mean in standard deviations; software today uses this cube-based version.

    G_1 = \dfrac{n}{(n - 1)(n - 2)} \sum z_i^3 =\dfrac{120}{119 \times 118} \times 132.09

    = 1.13

    Each student's z_i cubed adds up to \sum z_i^3 = 132.09, mostly from the heavy users. A plain average of the 120 cubes would be 1.10. The fraction in front, 120 ÷ 14,042, is about 1 ÷ 117 instead, which nudges that average up to make up for a small sample, giving 1.13, a clear lean toward high scores.

  8. 8. Measure How Heavy the Tails Are

    Raising z-scores to the fourth power makes the most extreme scores count even more. A normal curve averages 3 on this scale, so subtracting 3, scaled by the same small-sample adjustment as the first term, leaves how much more prone the scores are to extremes than normal.

    G_2 = \dfrac{n(n + 1)}{(n - 1)(n - 2)(n - 3)} \sum z_i^4 - \dfrac{3(n - 1)^2}{(n - 2)(n - 3)} =\dfrac{120 \times 121}{119 \times 118 \times 117} \times 522.51 - \dfrac{3 \times 119^2}{118 \times 117}

    = 1.54

    \sum z_i^4 is 522.51. A plain average over 120 students would be 4.35; the first fraction, the same kind of small-sample adjustment, scales it to 4.62 instead. The second fraction, 3.08, is a normal curve's 3, scaled the same way for a sample of 120. 4.62 − 3.08 = 1.54.

Run It in Software

The code below reads the running example's data from data.csv; save it in the same folder you run the code from.

SPSS (version 30, as documented by IBM)
  1. Analyze → Descriptive Statistics → Frequencies.
  2. Move screen_time_hours into Variable(s), and uncheck Display frequency tables.
  3. Click Statistics. Check Quartiles; Mean, Median and Mode; Std. deviation, Variance, Range, Minimum and Maximum; Skewness and Kurtosis. Click Continue.
  4. Click Charts, choose Histograms, click Continue, then OK.

Or paste this syntax:

FREQUENCIES VARIABLES=screen_time_hours
  /FORMAT=NOTABLE
  /NTILES=4
  /STATISTICS=MEAN MEDIAN MODE STDDEV VARIANCE RANGE MINIMUM MAXIMUM SKEWNESS SESKEW KURTOSIS SEKURT
  /HISTOGRAM.

Reading the Output

Look at the histogram first. Then read the Statistics table from the center out: Mean and Median, then Std. Deviation and the Percentiles 25 and 75 (their difference is the IQR), then Skewness and Kurtosis. SPSS's skewness and kurtosis are the sample-adjusted G₁ and excess G₂.

Statistics (screen_time_hours)
N Valid 120
N Missing 0
Mean 4.500
Median 4.100
Mode 3.4
Std. Deviation 1.9996
Variance 3.998
Skewness 1.129
Std. Error of Skewness .221
Kurtosis 1.541
Std. Error of Kurtosis .438
Range 10.7
Minimum 1.5
Maximum 12.2
Percentiles 25 3.000
Percentiles 50 4.100
Percentiles 75 5.600

The mode, 3.4 hours, is the single most common score (6 students), which says little with scores measured to a tenth of an hour. Analyze → Descriptive Statistics → Explore gives the same numbers plus a 95% CI for the mean, the IQR itself and a box plot.

Output laid out from this page's computed results, not captured from a live run; labels and layout may differ slightly in your version.

R 
screen <- read.csv("data.csv")
x <- screen$screen_time_hours

# length() counts missing values too; with any NAs, use sum(!is.na(x)) and na.rm = TRUE
length(x); mean(x); median(x); sd(x); var(x)
summary(x)
IQR(x)

# Skewness and kurtosis as SPSS reports them (G1, excess G2): type = 2
library(e1071)   # install.packages("e1071") the first time
skewness(x, type = 2)
kurtosis(x, type = 2)

# The heaviest user's z-score
(max(x) - mean(x)) / sd(x)

[1] 120
[1] 4.5
[1] 4.1
[1] 1.99958
[1] 3.998319
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
    1.5     3.0     4.1     4.5     5.6    12.2
[1] 2.6
[1] 1.128826
[1] 1.540742
[1] 3.850809

e1071's default is type = 3, a different formula that gives slightly smaller values (1.101 and 1.354 here), so set type = 2 to match SPSS (Joanes & Gill, 1998, compare the three versions). R's quartiles use a different rule from SPSS's (type 7, where SPSS uses type 6; Hyndman & Fan, 1996, Section 4); here they agree because the scores at both quartiles are tied.

Output from a real run: R 4.6.1 (e1071 1.7.17), run 2026-09-29.

Python 
import pandas as pd
from scipy import stats

# Run these lines one at a time in Jupyter or the Python prompt, which show each result;
# in a script, wrap each one in print().
screen = pd.read_csv("data.csv")

x = screen["screen_time_hours"]
x.describe()
x.median(), x.var()
x.quantile(0.75) - x.quantile(0.25)     # IQR

# G1 and excess G2, as SPSS reports them
stats.skew(x, bias=False)       # 1.128826
stats.kurtosis(x, bias=False)   # 1.540742

count    120.00000
mean       4.50000
std        1.99958
min        1.50000
25%        3.00000
50%        4.10000
75%        5.60000
max       12.20000
Name: screen_time_hours, dtype: float64

x.median() is 4.1, x.var() 3.998319 and the IQR 2.6. scipy's defaults (bias=True) give the unadjusted 1.115 and 1.428; bias=False matches SPSS. pandas' own x.skew() and x.kurt() already use the adjusted formulas.

Output from a real run: Python 3.9.6 (scipy 1.13.1, pandas 2.3.3), run 2026-09-29.

Impact

Effect Size

Descriptive statistics have no effect size of their own: nothing is being compared, so there's no effect. They're what effect sizes are built from. Cohen's d, for example, is a difference between two means divided by a standard deviation. The z-score is the same idea for a single person: a distance from the mean, in standard deviations.

z = \dfrac{x_i - \bar{x}}{s} = \dfrac{12.2 - 4.50}{2.00} = 3.85

The heaviest user's 12.2 hours is 7.70 hours above the average student, which is 3.85 of the usual spread between students. Every other student is within 3 SDs.

Whether 4.5 hours a day, or 12.2, is a problem is a question about students' sleep, study and moods, not statistics. The numbers say how much and how varied; the judgment comes from research on how screen time relates to those outcomes, and how strong that evidence is.

Uncertainty

There's no p-value here, because nothing is being tested. Descriptive statistics describe the sample in hand. To say how precisely the sample mean estimates the mean of everyone like these students, give a 95% confidence interval for the mean, built from the standard error. The SD says how much students vary; the standard error says how much the mean would shift from one sample of 120 to the next.

The standard error is 2.00 ÷ √120 = 0.18 hours. A 95% confidence interval reaches about two standard errors either side of the mean. The exact multiplier comes from the t distribution: 1.98 for 120 students (1.96 in very large samples). That's 4.50 ± 1.98 × 0.18 = 4.50 ± 0.36, so the interval for the mean of the population these students represent runs from 4.14 to 4.86 hours. It allows for chance in which students ended up in the sample, not for how they were recruited: if heavy users were more (or less) likely to volunteer, the interval can't correct for that. Like any t interval, it also assumes the students' scores are independent, with no clusters such as friends recruited together; with 120 students, the right skew matters little.

Testing whether that mean differs from a particular value, such as a guideline, is a one-sample t-test. SPSS also prints standard errors for skewness and kurtosis; dividing by them gives a rough test of shape, but in large samples it flags trivial departures, so look at the histogram instead.

Precision and Sample Size

Power is the chance that a study detects an effect that's really there. Descriptive statistics don't test for an effect, so there's no power to plan. What you can plan is precision: how narrow you need the confidence interval for the mean to be.

Assuming an SD of about 2 hours and students sampled independently, our 120 students give a 95% confidence interval of the mean ± 0.36 hours. To narrow it to ±0.25 hours (15 minutes), we'd need about (1.96 × 2 ÷ 0.25)² ≈ 246 students, where 1.96 is the normal curve's value for 95%; using the t value in place of 1.96, as the interval itself does, makes it 249.

The interval narrows with the square root of the sample size: to halve its width, collect about four times as many people. The planner below starts at our SD of 2 hours and a target of ±0.25 hours. Try it:

Other Info

Follow-Up and Post Hoc Tests

None. This page estimates rather than tests, so there are no follow-up comparisons to run.

References

  • Pearson, K. (1894). Contributions to the mathematical theory of evolution. Philosophical Transactions of the Royal Society of London. A, 185, 71–110. https://doi.org/10.1098/rsta.1894.0003
  • Pearson, K. (1895). Contributions to the mathematical theory of evolution. II. Skew variation in homogeneous material. Philosophical Transactions of the Royal Society of London. A, 186, 343–414. https://doi.org/10.1098/rsta.1895.0010
    “it seems to me better to take as the measure of skewness the ratio of the distance between the maximum ordinate and the centroid to the length of the swing radius of the curve about the centroid vertical” (Section 14, p. 370)
  • Hyndman, R. J., & Fan, Y. (1996). Sample quantiles in statistical packages. The American Statistician, 50(4), 361–365. https://doi.org/10.1080/00031305.1996.10473566
    “In this section we summarize the sample quantile definitions that are implemented in some major statistical packages.” (Section 4, p. 364)
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  • Westfall, P. H. (2014). Kurtosis as peakedness, 1905–2014. R.I.P. The American Statistician, 68(3), 191–195. https://doi.org/10.1080/00031305.2014.917055
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