Measures of Central Tendency Quiz

SDCE 303: Introduction to Biostatistics

• Lecture Notes 4

Descriptive Statistics: Measures of Central Tendency & Dispersion

• Objectives:
  • Explain common averages (Mode, Median, Mean)
  • Explain common measures of variation (Range, Variance, Standard deviation, Coefficient of variation)

Measures of Central Tendency

• Measures of central tendency summarize a data set by locating its center.
• A measure of central tendency indicates the middle of the data.
• The three most common measures are Mode, Median, and Mean.

Mode

• The mode is the value that appears most frequently in a data set.
• Data can have one mode (unimodal), two modes (bimodal), or more than two modes (multimodal).
• Some data sets have no mode (if all values are unique).
• The mode is not unique.

Mode from Ungrouped Data

• Arrange data in an array.
• Identify the value(s) occurring most frequently.
• Example: 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10, 11, 11, 14, 14, 14, 14, 14
  • Mode = 8 and 14 (bimodal)

Mode from a Simple Frequency Distribution

• For discrete distributions, the mode is the value with the greatest frequency.
• Example:

  Score	No. of students
  30	1
  24	4
  12	4
  18	3
  22	1
  20	2
  14	2
  15	1

  • Mode = 24 and 12 (bimodal)

Mode from the Histogram

• Draw the histogram from the frequency distribution.
• Use the highest modal class and find the intersection of the lines on the histogram.
• If the modal class is at the extreme group, take the mean of the lower and upper boundaries.

Using Formula to Determine Mode for Grouped Data

• Mode = L + D1/(D1 + D2) x C
  • L = lower boundary of the modal class
  • D1 = difference between the largest frequency and the frequency immediately before it
  • D2 = difference between the largest frequency and the frequency immediately after it
  • C = modal class width

Median

• The median is the middle value in a sorted dataset.
• Arrange the data in ascending order.
• If the number of data points is odd, the median is the middle value.
• If the number of data points is even, the median is the average of the two middle values.

Median from a Frequency Table

• If n is odd, the median is the value corresponding to (∑f+1) /2 th item (Total frequency + 1 divided by 2).
• If n is even, the median is the mean of the (∑f) /2 th and (∑f +1)/2 th items.
• Add frequencies until you get to the median position item.

Median from Cumulative Curve

• From the cumulative curve, the median is the value on the x-axis corresponding to half the total frequency.
• Draw the cumulative curve using the absolute cumulative or % cumulative values.
• The median is the 50% mark traced to the x-axis using the absolute values; or the 50% mark using the % values.

Properties of the Median

• Uniqueness: For a given set of data there is only one median.
• Not affected by extreme values.
• Generally, the median provides a better measure of location than the mean when there are some extremely large or small values.

Mean

• The mean is the sum of all values divided by the total number of values.
• Population mean: μ= Σ xi /N
• Sample mean: X̄ = Σ xi /n

Sample Mean

• If the population is large, use the sample mean to approximate the population mean.
• Sample mean is a statistic (characteristic of the sample)

Mean of a Frequency Distribution

• Ungrouped data: Mean = ∑x /∑f
• Grouped data: Mean = ∑fx /∑f, where x is the class mid-point value

Weighted Arithmetic Mean

• When values are not equally important, assign weights.
• Weighted mean = [ Σ(weights * values)] / Σ(weights)

Other Forms of the Mean

• Arithmetic mean
• Geometric mean
• Harmonic mean
• Trimmed mean

Uses of the Mean

• Statistical work (e.g., correlation, ANOVA)
• Normal distribution data
• Situations demanding an average

Properties of the Mean

• Affected by every score
• Mean is a function of sum of scores
• Sum of the deviations from the mean is 0

Skewness

• Symmetrical: mean = median = mode
• Positively skewed: mean > median > mode
• Negatively skewed: mean < median < mode

Measures of Shape (Symmetric or Skewed):

• A distribution can be characterized as symmetric, left-skewed (negatively skewed), or right-skewed (positively skewed), depending on the spread of the values.
• The direction of skewness is determined by comparing the mean, median, and mode.

Variation

• Describes how widely the data is spread around the center
• Measures of variation:
  • Range
  • Variance
  • Standard Deviation
  • Coefficient of Variation

The Range

• The difference between the largest and smallest values in the data

Variance

• Measures how spread out the values are relative to the mean.
• Sample variance: S^2 = Σ(xi - X̄)^2 / (n - 1)
• Population variance: σ² = Σ(xi - μ)² / N

Standard Deviation

• The square root of the variance.
• Provides a measure of dispersion in the same unit of measurement as the observations.
• A lower standard deviation indicates less dispersion.

Steps for Computing Variance & Standard Deviation

1. Calculate the mean.
2. Calculate (x-mean).
3. Square each (x-mean)^2.
4. Sum the results from step 3.
5. Divide the sum by (n-1) for sample variance.
6. Find the square root of the variance to obtain the standard deviation.

Coefficient of Variation

• Measures relative dispersion in two independent data sets.
• C.V = S/ X * 100%, where X = sample mean and S = sample standard deviation.

Normal Distribution

• A common type of probability distribution that results in a bell-shaped curve when graphed.
• The mean, median, and mode are all equal.
• Characterized by its symmetrical shape and how data is spread out from its mean value

Z-Scores

• Standardize data to compare to the normal distribution
• Allow proportional conversion of a study sample to the whole population.
• Shows the number of standard deviations a value is away from the mean.
• Calculated as: Z = (x-μ)/ σ ; where x is the data point, and μ is the mean and σ is the standard deviation of the population.

Using Z-Scores to Standardize a Distribution

• The mean remains 0.
• Standard deviation remains 1.

Test your knowledge on the concepts of central tendency, including mode, median, and their calculations. This quiz covers definitions, properties, and methods for finding the mode and median in various datasets. Challenge yourself with questions designed to reinforce your understanding of these statistical measures.