Measures of Central Tendency

PSY 348: Lecture 3

Dave Brocker

Farmingdale State College

Review

Scales of Measurement

  • What are the 4 scales of measurement

  • What are the only scales of measurement that matter for us?

Review

Scales of Measurement

  • Nominal

  • Ordinal

  • Interval

  • Ratio

Review

Statistical Terms

  • n = ?

  • x = ?

Review

Statistical Terms

  • n = the number of people in the sample

  • x = one participant’s score

Review

Statistical Terms

  • What is a parameter?

  • What is a statistic?

Review

Statistical Terms

  • What is a parameter?

    • Information that describes the population.

  • What is a statistic?

    • Information that describes the sample.

    • Inferences about the population based on information from the sample.

Descriptive Statistics

Describe the characteristics of the sample.

  • Think about our data on time spent on Instagram.

  • How could we describe it?

    • Average

    • Min & Max

Descriptive Statistics

Describe the characteristics of a sample in terms of:

  • Central tendency: What most people said

  • Dispersion: How spread out the data are

Measures of Central tendency

  • Tell us about the mid-point (center) of a distribution.

  • Tell us about the mid-point (center) of a distribution.

  • Tell us most participants’ answer.

Measures of Central tendency

  • Mean

    • the average; obtained by summing all values and dividing by the number of values.

    • \(\sum^{n}_{i=1} = x_1 + x_2...\)

Measures of Central tendency

  • Median

    • The middle number; obtained by ordering all values from lowest to highest and taking the middle (if n is odd or the average of the 2 middle if n is even)

Measures of Central tendency

  • Mode

    • the most frequent answer; obtained by counting how many times each answer is given and taking the value that occurs most often

Measures of Central tendency

Standard Normal Distribution

Measures of Central tendency

In a normal distribution, the mean, median, and mode are equal to one another.

Mean = Median = Mode

Measures of Central tendency

The Mean is good for use in normal distributions

The Mean

  • The mean is only helpful in a normal distribution, which you may have heard called a normal curve or bell curve.

  • We overwhelmingly use the mean, because in the social and behavioral sciences, we nearly always assume the distribution is normal.

Measures of Central tendency

When should we use the median…

A skewed distribution occurs when one side of the date gets cut off due to measurement limitation.

The Median

  • In a skewed distribution, the mean gets pulled out toward the tail, and the mode gets pulled to the cluster.

  • The median is the best measure of central tendency for skewed distributions because it tells us how most people answered.

Measures of Central tendency

The Mode

  • The mode is good for bi-or-tri-modal distributions.

  • When might you have a bi-modal distribution?

Measures of Central tendency

The Mode

Measures of Central tendency

The Mode

When should we use…

  • Mean

    • good for normal distributions

  • Median

    • good for skewed distributions

  • Mode

    • good for bimodal distributions

Central tendency: Which should we use?

Two or More Peaks

Central tendency: Which should we use?

Slide to the Left!

Central tendency: Which should we use?

Looks Normal to Me

Measures of Central tendency:

The mean

  • We use the mean most often in social science, because many of our statistical tests can only be used on normal distributions.

  • We assume normality of the distribution, and use the mean.

Calculating measures of central tendency

Calculating The mean

  • Sum all x values

  • Divide by the number of x values (n).

. . .

x = 4,2,5,6

. . .

\(\frac{\Sigma(x)}{n}\)

. . .

\(\Sigma(x) = 4 + 2 + 5 + 6 = 17\)

. . .

\(\frac{17}{4} = 4.25\)

Calculating measures of central tendency

Calculating The median

  • Put all x values in order from smallest or largest or largest to smallest.

  • Find the middle number.

  • If there are an even number of x values, take the average of the 2 middle numbers.

  • x = 4,2,5,6,3

  • x = 2,3,4,5,6

Calculating measures of central tendency

Calculating The mode

  • Put all x values in order from smallest or largest or largest to smallest. \(x = [3,2,1,5,3,7,8,3,2,4,1]\)

  • Find the number or numbers that repeat the most. \(x = [1,1,2,2,3,3,3,4,5,7,8]\)

  • There can be more than 1 mode.

Using Excel

Excel Practice

  • Cell

  • Columns

  • Rows

Excel Practice

  • What does each row in this spreadsheet represent?

  • What does each column in this spreadsheet represent?

Excel Practice

  • Adding & Summing

  • Subtracting

  • Multiplying

  • Dividing

  • Squaring

Excel Practice

  • Adding & Sum: SUM(B2:B10)

  • Subtracting: C9 - C2

  • Multiplying: B12 * 4

  • Dividing: A2/B3

  • Squaring: (B2-C2)^2

  • Average: AVERAGE(B2:B20)

Excel Practice

  • Open our Class Data in Microsoft Excel.

  • Calculate the Mean

  • Using (SUM(x))/n

  • Using Average(x)

  • Calculate the Median

  • Find the Mode(s)

Review

When do we use the…

  • Mean?

  • Median?

  • Mode?

Review

How do we calculate the…

  • Mean?

  • Median?

  • Mode?

Review

What is this the formula for?

  • Explain this formula to me.

. . .

\(\sum(x)\)

Review

When are the mean, median, and mode the same (equal to one another)?