Measures of Central Tendency

Measures of Central Tendency

• The three main measures of central tendency are the mean, median, and mode.^1,2

Mean

• The mean is often referred to as the arithmetic average, and it is symbolized as x̄.¹
  • It is the most widely used measure of central tendency.^1,2
• It is calculated by adding all the values in a dataset and dividing the sum by the total number of observations.^1,2
• Advantages:
  • The mean considers all values in the dataset making it very comprehensive.¹
  • It is stable across repeated samples from the same population.¹
  • It is a unique value meaning that there is only one mean for a given dataset, making it useful when comparing different groups.²
  • It utilizes every observation in the dataset.²
• Disadvantages:
  • It is sensitive to outliers, especially when the dataset is small.^1.2
  • It cannot be used for categorical or ordinal data.¹
• The weighted mean can be used to reduce the effect of outliers:¹
  • Each value is multiplied by a weight which is based on frequency or importance.¹
  • Weighted values are summed and divided by the total weights.¹
  • Outliers have less influence since their weight is small.¹
• Example
  • During induction of general anesthesia, an anesthesiologist records the heart rate of 5 patients one minute after propofol administration:
    • 72, 78, 81, 75, 84 beats per minute
    • To calculate the mean heart rate (x̄):
      • x̄=(72+78+81+75+84)/5
      • x̄=78 bpm
  • The mean heart rate for these patients after induction was 78 bpm.
    • This provides a single representative value that summarizes the group’s response.

Median

• The median is the midpoint of a dataset after the values are arranged in order from either ascending or descending value.²
• It divides the dataset into two equal halves, with 50% of values lying below and 50% above it.¹
  • If the dataset has an odd number of values, the median is the middle observation.¹
  • If the dataset has an even number of values, it is determined from the average of the middle two observations.¹
• Advantages:
  • It is easy to calculate and understand.¹
  • It works with ratio, interval, and ordinal data.¹
  • It is not affected by extreme values, making it suitable for skewed datasets.²
  • Similar to the mean, the median is also a unique value, making it useful for group comparisons.²
• Disadvantages:
  • It does not use all values, only the position in the ordered dataset.¹
  • It cannot be easily used in statistical models.¹
  • It is less popular than the mean, even though it is often more robust in the presence of outliers.²
• Example
  • An anesthesiologist records the time to eye opening after extubating in 7 patients: 6, 8, 7, 12, 9, 15, 10 minutes
  • Step 1: Arrange values in ascending order:
    • 6, 7, 8, 9, 10, 12, 15
  • Step 2: Find the middle value
    • Since there are 7 values, the median is the 4th observation.
  • Median = 9 minutes
    • This means that half the patients woke up in less than 9 minutes, and half took more than 9 minutes.

Mode

• The mode represents the most frequently occurring value within a dataset.^1,2
• Datasets can be bimodal or multimodal.^1,2
• Advantages:
  • It identifies the most frequent value in a dataset.^1,2
  • It is useful for describing bimodal or multimodal distributions (two or more peaks).¹
  • It is the only measure that can be used with nominal data.¹
• Disadvantages:
  • It is strongly influenced by small sample sizes.¹
  • It is rarely used in advanced statistical analysis.¹
  • Since multiple modes can exist in a single dataset, it is not reliable for comparing between groups.²
• Example:
  • An anesthesiologist measures the time to loss of consciousness after propofol administration in 9 patients: 40, 35, 38, 42, 40, 37, 40, 36, 38 seconds
  • Step 1: Count the frequency of each time:
    • 35 sec: 1
    • 36 sec: 1
    • 37 sec: 1
    • 38 sec: 2
    • 40 sec: 3
    • 42 sec: 1
  • Step 2: Identify the most frequent value:
    • Mode = 40 seconds
  • The most common time to loss of consciousness in this group was 40 seconds, making it the mode of the dataset.

Impact of Distribution

• The mean, median, and mode of a dataset reflect the shape of the distribution.¹

Normal Distribution

• The bell-shaped normal distribution is the most frequently encountered statistical distribution.³
  • Numerous inferential statistical tests rely on the assumption that datasets conform to this distribution.³
• In a normal distribution, the mean, median, and mode all align at the same central point (Figure 1).^1-3

Skewed Distribution

• The mean shifts toward the tail of the distribution curve as it is influenced by extreme values.^1,2,4
• The mode remains at the peak.^1,4
• The median typically falls between the mean and mode.^1,4

Positive skew (right-skewed) (Figure 1):

• The tail is longer on the right side.¹
• The bulk of values lie to the left of the mean.¹

Negative skew (left-skewed) (Figure 1):

• The tail is longer on the left side.¹
• The bulk of values lies to the right of the mean.¹

Sample Size and Central Tendency

• Sample size plays a critical role in how accurate the measures of central tendency are.^1,5
• Sample size directly influences variability.¹
  • Larger sample sizes reduce standard deviation, concentrating more data around the mean.¹
  • Smaller samples increase standard deviation.^1,2
    • Small samples also reduce reproducibility, as chance plays a larger role in shaping the distribution.¹
• The mean is more affected by variability and outliers when sample sizes are small, making it less reliable.¹
• The median is often more stable under these conditions.¹
• In clinical research, the choice between mean and median must account for both the data’s shape and the sample size of the study.¹