Measures of center and measures of spread



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  • The following topics will be covered in this lecture:
    • Mean
    • Median
    • Mode
    • Midrange
    • Computing the mean from frequency distributions
    • Weighted means
    • Basic concepts of variation

Characteristics of data

Diagram of the percent of outcomes contained within each standard deviation of the mean
for a standard normal distribution.

Courtesy of M. W. Toews CC via Wikimedia Commons.

  • Recall, we try to characterize data by a number of the features that it exhibits.
  • Some of the key measures are:
    1. Center: A representative value that indicates where the middle of the data set is located.
    2. Variation: A measure of the amount that the data values vary.
    3. Distribution: The nature or shape of the spread of the data over the range of values (such as bell-shaped).
    4. Outliers: Sample values that lie very far away from the vast majority of the other sample values.
    5. Time: Any change in the characteristics of the data over time.
  • We will now begin studying measures of center.
  • There are several main measures of center of a data set:
    1. mean;
    2. median;
    3. mode; and
    4. midrange.
  • Each of these usually gives a different view of where the “most central point” of the data lies.


  • The (arithmetic sample) mean is usually the most important measure of center.

  • Suppose we have \( n \) total sample measurements of some variable \( x \).

    • We will denote these samples \( x_1, x_2, \cdots, x_n \)
  • Then, the (arithmetic sample) mean is defined

    \[ \text{Sample mean} = \frac{x_1 +x_2 +\cdots + x_n}{n}= \frac{\sum_{i=1}^n x_i}{n} \]

  • Discuss with a neighbor: is the sample mean a statistic or a parameter?

    • A: the sample mean is computed from samples and thus a statistic.
    • For this reason, if we took new measurements from a new sample of the population, we could get a different value.
    • The random difference between the sample mean and the mean of the true population mean is called sampling error.
  • An important property of the sample mean is that it tends to vary less over re-sampling than other statistics.

    • That is, it tends to stay close to the same value.
  • However, the sample mean is very sensitive to outliers.

    • If outliers exist in the data, the mean can be drawn far away from the “main” cluster of data.
  • A statistic is called resistant if it doesn't change very much with respect to outlier data.


  • A different notion of center is the middle of the data.
  • For a numerical measurement, we can always order the data so that we go from low to high or high to low.
  • Median – the median is the middle of the ordered data set.
    • If there are an odd number of samples, the median is defined as the middle value exactly.
    • If there are an even number of samples, we split the data into the lower \( 50\% \) and upper \( 50\% \) of the samples;
    • then we take the median to be the mean of the:
      1. largest of the lower \( 50\% \); and
      2. smallest of the upper \( 50\% \).
  • Suppose we are given a list of the following samples \( 22, 22, 26, 24, 23 \).
    • Discuss with a neighbor: what is the median of this list of samples?
    • Ordering the values, we get \( 22, 22, 23, 24, 26 \) so that the middle value is obviously \( 23 \).
  • Suppose a new sample includes \( 22, 22, 26, 24, 23, 27 \).
    • Discuss with a neighbor: what is the median of this list of samples?
    • In this case, we have an even number of samples.
    • The lower \( 50\% \) is given by \( 22,22,23 \) and the upper \( 50\% \) is given by \( 24,26,27 \).
    • Therefore, the mean of the largest lower value and the smallest upper value is given by \[ \frac{23 + 24}{2} = 23.5. \]

Median continued

  • Let us consider the last example once again.

  • Suppose our sample includes the values \( 22, 22, 26, 24, 23, 27 \).

  • If we compute the (arithmetic sample) mean, we find

    \[ \frac{22+22+26+24+23+27}{6} = \frac{144}{6} = 24. \]

  • Now, suppose that we realize that the value \( 27 \) was obtained due to measurement error and our sample should have read \( 22, 22, 26, 24, 23, 1000 \).

  • Discuss with a neighbor: by replacing the value \( 27 \) with \( 1000 \) does this affect the median? Does this affect the mean? Which of these statistics are resistant to outliers?

    • We note, this does not affect the median – indeed the actual numerical value of the final measurement does not change which value lies in the middle.
    • The lower \( 50\% \) of the measurements are given by \( 22,22,23 \) and the upper \( 50\% \) are given by \( 24,26,1000 \).
    • Once again, we compute the mean of the largest lower value and the smallest upper value, given by \( \frac{23 + 24}{2} = 23.5. \) so that we say the median is resistant to outliers.
    • On the other hand, the sample mean is given as \[ \frac{22+22+26+24+23+1000}{6} = \frac{1117}{6} \approx 186.1667. \]


  • Another notion of the most “central point” in the data can be the value that is measured most frequently.

  • Mode – the mode is the observed value that is most frequent in the data.

  • Consider the last example with samples of \( 22, 22, 26, 24, 23, 27 \). Q: What is the mode?

    • In this case, we sampled \( 22 \) more than any other value, so this is the mode of the data.
  • When two or more values have the highest frequency, we call the data bi-modal or multi-modal.

    • An exception to this above rule is when no values are repeated.
    • In this case, we say there is no mode to the data.

Differences in mean, median and mode