MC³ LabCUHK Psychology

Lesson 10 · Statistics

Describe data

Report the middle and the spread of a variable, overall and for each group.

In this lesson

  • Mean, median, SD and IQR
  • One summary table per group
  • Everything at once with describe()

You need

Datastudents.csvStudent Life Survey: 300 students, with their study habits, sleep, stress, wellbeing and exam results.
Scriptlesson-10.RAll the code in this lesson, ready to run.

Set up

R code
library(tidyverse)
students <- read_csv("data/students.csv", show_col_types = FALSE)

Descriptive statistics answer two questions: where is the middle? and how spread out are the values?

The middle: mean or median?

Six work-hour values on a line: five between 38 and 45 and one at 90. The median sits among the five; the mean is pulled towards 90

One extreme value pulls the mean (average) but not the median (middle value). Use the median when data are lopsided or have extreme values.

R code
mean(students$Sleep, na.rm = TRUE)
median(students$Sleep, na.rm = TRUE)
Output
[1] 6.412414
[1] 6.5

The spread

R code
sd(students$Sleep, na.rm = TRUE)
min(students$Sleep, na.rm = TRUE)
max(students$Sleep, na.rm = TRUE)
quantile(students$Sleep, probs = c(0.25, 0.75), na.rm = TRUE)
IQR(students$Sleep, na.rm = TRUE)
Output
[1] 0.8189691
[1] 4
[1] 9
25% 75% 
5.9 6.9 
[1] 1
  • SD (standard deviation): the typical distance from the mean. Students sleep about 6.4 hours, give or take 0.8.
  • IQR (interquartile range): the spread of the middle half. Goes with the median.
  • Report mean + SD for symmetrical data, median + IQR for lopsided data.

One table per group

R code
students %>%
  group_by(Gender) %>%
  summarise(n = n(),
            missing = sum(is.na(Sleep)),
            mean = mean(Sleep, na.rm = TRUE),
            sd = sd(Sleep, na.rm = TRUE),
            median = median(Sleep, na.rm = TRUE),
            IQR = IQR(Sleep, na.rm = TRUE))
Output
# A tibble: 2 × 7
  Gender     n missing  mean    sd median   IQR
  <chr>  <int>   <int> <dbl> <dbl>  <dbl> <dbl>
1 Female   184       7  6.39 0.820    6.4  1.10
2 Male     116       3  6.45 0.820    6.5  1.1

Women and men sleep about the same.

Everything at once: describe()

R code
library(psych)
students %>%
  select(Sleep, Stress, ExamScore) %>%
  describe()
Output

Attaching package: ‘psych’

The following objects are masked from ‘package:ggplot2’:

    %+%, alpha
          vars   n  mean    sd median trimmed   mad min max range
Sleep        1 290  6.41  0.82    6.5    6.42  0.74   4   9     5
Stress       2 300  5.94  1.85    6.0    5.97  1.48   2  10     8
ExamScore    3 300 67.48 13.05   67.0   67.44 11.86  34 100    66
           skew kurtosis   se
Sleep     -0.15    -0.09 0.05
Stress    -0.10    -0.55 0.11
ExamScore  0.03    -0.08 0.75

The key columns: n (non-missing), mean, sd, median, min, max, and skew (0 = symmetrical; between −1 and +1 is fine). The message when loading psych is normal.

Exercises

Exercise 10.1 Dataset: students.csv

Describe exam scores

Find the mean, median, SD and IQR of ExamScore. Is it roughly symmetrical?

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R code
mean(students$ExamScore)
median(students$ExamScore)
sd(students$ExamScore)
IQR(students$ExamScore)
Output
[1] 67.47667
[1] 67
[1] 13.05415
[1] 17

Mean (67.5) and median (67) are almost equal, so it's roughly symmetrical.

Exercise 10.2 Dataset: students.csv

Compare groups

For each Motivation group: number of students, mean and SD of ExamScore, rounded to 1 decimal place. Show the groups in the order Low, Medium, High.

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Make Motivation a factor first (Lesson 7), then group_by() and summarise(). Wrap with round(..., 1).

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R code
students %>%
  mutate(Motivation = factor(Motivation, levels = c("Low", "Medium", "High"))) %>%
  group_by(Motivation) %>%
  summarise(n = n(),
            mean = round(mean(ExamScore), 1),
            sd = round(sd(ExamScore), 1))
Output
# A tibble: 3 × 4
  Motivation     n  mean    sd
  <fct>      <int> <dbl> <dbl>
1 Low           66  58.7  11.7
2 Medium       147  67.7  12.2
3 High          87  73.7  11.7
Exercise 10.3 Dataset: students.csv

describe()

Use describe() on Age, StudyHours and Attendance. Which is the most skewed? Why might that be?

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R code
students %>% select(Age, StudyHours, Attendance) %>% describe()
Output
           vars   n  mean   sd median trimmed  mad min max range
Age           1 300 21.09 1.99     21   20.88 1.48  18  29    11
StudyHours    2 294 14.07 5.98     14   14.13 5.93   0  31    31
Attendance    3 300 83.46 9.38     84   83.77 8.90  56 100    44
            skew kurtosis   se
Age         0.98     0.91 0.11
StudyHours -0.03    -0.32 0.35
Attendance -0.36    -0.12 0.54

Age (skew 0.98): most students are 19 to 22, with a tail of a few older students.

Key points

  • Middle: mean (with SD) or median (with IQR).
  • Add na.rm = TRUE when there are missing values, and report n.
  • describe() gives a full summary in one line.