2.1.Displaying & Describing Data

Measures of central tendency and spread

Bárbara D. Bitarello

2025-09-15

Outline

• Estimates of location
• Estimates of width
• Explanatory and exploratory figures
• Best practices in figure design
• How data types drive figure design
• How to make effective tables

today: location and width; next class plots

Learning Goals

• Differentiate between estimates of location and estimates of width.
• Recognize that variability is not simply noise, but is a key parameter that can be estimated.
• Become familiar with the most common descriptive statistics.
• Know when the mean or median is a more appropriate summary of location.
• Distinguish between explanatory and exploratory figures.
• Identify what makes a good graph.
• Understand how data types drive figure design.
• Understand how to make effective tables.

These learning goals are for the entire topic/unit. They will also be available in the study guide.

Frequency vs. Probability Distribution

Frequency Distribution

• sample
• a snapshot of actual counts of outcomes in a given sample

Probably Distribution

• population
• usually not known so based on probability rules: it represents the theoretical probability of all possible outcomes of a random variable

Descriptive Statistics

Or summary statistics: quantities that capture important features of frequency distributions

Three Common Descriptions of Data

• Location (central tendency)

• Width (spread)

• Association (correlation)

Measures of location (central tendency)

Measures of location

• Mean: The weight of your data. The average value.

• Median: A “typical individual”. If I take an individual at random, this is the value we expect them to be closest to.

• Mode: The typical individual most common value for an individual. The most likely answer for an individual selected at random.

The average problem

“Average” is often used synonymously with mean.

• But: medians and even modes are sometimes called an “average”.

• When you see the word, “average”, pay attention to which measure of location it describes.

In this course, you should use statistical terms precisely, i.e., to indicate the same consistently. So if you want to say mean, say mean, not average.

(Arithmetic) Mean

The sum of values divided by the sample size.

\[\bar{Y}=\frac{\sum_{i=1}^{n}Y_{i}}{n}\]

• \(\bar{Y}\) is the mean value of variable \(Y\)
• \(\sum\) signified “sum”

• \(n\) is the number of individuals in the samples (sample size)
• \(Y_{i}\) is the observed value for the \(i\)-th individual

Example: The mean of the set of 11 numbers: \(1, 15,9, 16,6, 17, 10, 5, 12, 14, 13\) is

\(\bar{Y} = 1+15+9+16+6+17+10+5+12+14+13/11 = 106.1818\)

This is the “arithmetic mean”, the one you see most often. The three pytagorean means also include the geometric mean and the harmonic mean.

Practice: Mean from a frequency table

• Frequency tables show the number of times, \(n_{i}\) a value, \(Y_{i}\), is observed in a sample of size \(n_{total}\)

• We calculate the mean from a frequency table by summing the product of \(n_{i}\) and \(Y_{i}\) all values and diving by \(n_{total}\).

convictions	frequency
0	265
1	49
2	21
3	19
4	10
5	10
6	2
7	2
8	4
9	2
10	1
11	4
12	3
13	1
14	2

A frequency table. Data from Farrington (1994) and distributed at http://www.webapp.icpsr.umich.edu/cocoon/NACJD-STUDY/08488.xml.

Practice: Mean from a frequency table

• Frequency tables show the number of times, \(n_{i}\) a value, \(Y_{i}\), is observed in a sample of size \(n_{total}\)

convictions <- c(0, 1, 2, 3, 4,
    5, 6, 7, 8, 9, 10, 11, 12,
    13, 14)
freqs <- c(265, 49, 21, 19, 10,
    10, 2, 2, 4, 2, 1, 4, 3, 1,
    2)
n_total <- sum(freqs)  # 395

• We calculate the mean from a frequency table by summing the product of \(n_{i}\) and \(Y_{i}\) across all values and diving by \(n_{total}\).

# multiply two vectors of
# same length

FinalMean <- sum(convictions *
    freqs)/n_total
FinalMean

[1] 1.126582

A frequency table

convictions	frequency	ConvictionxFreq
0	265	0
1	49	49
2	21	42
3	19	57
4	10	40
5	10	50
6	2	12
7	2	14
8	4	32
9	2	18
10	1	10
11	4	44
12	3	36
13	1	13
14	2	28

Median

The value halfway through an ordered list of observations.

• The \((n + 1) / 2\)-th value for odd sized samples.
• Mean of n/2 th and the \((n + 2) / 2\)-th value for even sized samples.

Example: Using the same set of numbers as before (try):

• First, order the numbers: \(1, 5, 6, 9, 10, 12, 13, 14, 15, 16, 17\)
• The \((n + 1) / 2\)-th value for odd sized samples: \(12\)

Mean and Median in R

# a vector with the numbers
mynumbers <- c(1, 15, 9, 16, 6, 17, 10, 5, 12, 14, 13)
# mean 'manually'
mysum <- sum(mynumbers)  #sum all numbers
l <- length(mynumbers)  #number of elements in vector
mymean <- mysum/l
mymean

[1] 10.72727

# note that you can use a built-in function for this:
mymean2 <- mean(mynumbers)
mymean2

[1] 10.72727

# median with built-in function
mymedian <- median(mynumbers)
mymedian

[1] 12

# median 'manually'
mynumbers2 <- sort(mynumbers)
mynumbers2

 [1]  1  5  6  9 10 12 13 14 15 16 17

# since l = 11 (odd) we select the l+1-th element, i.e, (11+1)/2=6
mynumbers2[6]

[1] 12

That’s all for today

From: makeameme.org