Central Tendency and Measures of Variance

Central Tendency and Measures of Variance

Introduction

• This lecture combines chapters two and three, focusing on central tendency and measures of variance.

• The goal is to ensure a basic understanding of mean, median, mode, range, and standard deviation for the exam covering chapters one through four.

• Descriptive statistics, frequency distribution tables, and central tendency should be approached with time and precision.

Central Tendency

Definition

• Central tendency involves using a single number to describe a large dataset.

• Example: The average cost of a car purchased in Southern California between January and June.

• Averages are commonly used, such as GPA, which represents all grades in an undergraduate degree with one number.

Purpose

• To find a number that represents the middle of all collected numbers.

Mean

• The most common measure of central tendency.

• It does not indicate how scores vary or spread out (variability).

• The mean is the balance point of the scores and is sensitive to extreme scores.

• Suitable for interval data.

Formula

• The mean is calculated as the sum of all scores divided by the number of scores.

  • For a sample:

    • $M = \frac{\sum X}{n}$

    • Where:

      • $M$ = sample mean

      • $\sum X$ = sum of all raw scores

      • $n$ = total number of scores in the sample

  • For a population:

    • $μ = \frac{\sum X}{N}$

    • Where:

      • $μ$ = population mean

      • $\sum X$ = sum of all raw scores

      • $N$ = total number of scores in the population

Example Calculation

• Linda's exam scores: 58, 67, 60, 84, 93, 98, 100.

• Calculate the mean: $(58 + 67 + 60 + 84 + 93 + 98 + 100) / 7 = 560 / 7 = 80$

• Linda's grade in the class is a B (80%).

Computing the Mean from a Frequency Table

• Formula: $mean = \frac{\sum(f ")}{n}$

  • Multiply each $x$ (score) by its frequency $(f)$.

  • Sum the products of each frequency multiplied by the raw score.

  • Divide by the total number of scores.

  • Example:

    • Scores: 6 (frequency 1), 5 (frequency 0), 4 (frequency 3), 3 (frequency 3), 2 (frequency 2)

    • $(6<em>1) + (5</em>0) + (4<em>3) + (3</em>3) + (2*2) = 6 + 0 + 12 + 9 + 4 = 31$

    • Total number of scores = 9

    • Mean = $31 / 9 = 3.44$

When Not to Use the Mean

• The mean is susceptible to outliers (extreme scores that lie outside the trend).

  • Outliers can skew the mean, making it a less representative measure of central tendency.

• Example: Salaries of 10 staff members where two large salaries skew the mean.

  • Most workers earn between $12,000 and $18,000, but the mean salary is $30,700 due to the manager's salaries.

  • In skewed situations, the median is a better measure of central tendency.

Median

• The score that marks the 50th percentile, with half of the scores above and half below.

• The median is the middle score or middle value.

• Uses position rather than the actual value of the data.

• Changes in extreme values do not affect the median.

• Applicable to interval data.

Steps to Find the Median

• Put the scores in rank order (smallest to largest).

• For an odd number of data points, the median is the middle value.

• For an even number of data points, add the two middle numbers and divide by two.

Examples

• Even set: 10, 12, 19, 25, 28, 30. Middle numbers are 19 and 25. Median = $(19 + 25) / 2 = 22$

• Odd set: 10, 12, 19, 25, 28. The median is 19.

• For large datasets, software like SPSS can be used to find the median.

Mode

• The data point that occurs most frequently.

• No calculations are needed; it is found by inspection.

• There can be more than one mode in a dataset.

• Best used with nominal data (categorical data collected by name).

Example

• Count the letters in each word of the sentence: "Count the letters in each word of the sentence and then find the mode."

• Counts: 5, 3, 7, 2, 4, 4, 2, 4, 1, 2, 3, 4, 5, 3, 4.

• The mode is 4, as it appears most frequently (five times).

Data Skewness

• Skewed data means the data doesn't have an equal distribution.

• In a normal distribution, the mean, median, and mode are all in the center.

• If data is skewed (not a normal shape), the mean is pulled in the direction of the skew.

• Right-skewed data: the tail is on the right side.

  • Mode is at the peak, the median is to the right of the mode, and the mean is pulled towards the right (the tail).

  • In this case, the mean does not represent the dataset well; the median is a better measure.

• When data is skewed, the median is a more appropriate measure of central tendency.

Variability

Definition

• Variability provides a quantitative measure of how spread out the scores are in a distribution.

• Small differences: variability is small.

• Large differences: variability is large.

• Variability describes the distribution by how far the values deviate from the average.

Range

• The difference between the largest and smallest scores.

• Measures the spread of the data but only uses the highest and lowest values, ignoring the other values.

• Example: John's scores range from 75 to 99 (range = 24), while Joe's range from 80 to 90 (range = 10).

• Smaller range indicates more consistency.

Standard Deviation

• Measures the dispersion of data values around the mean (how much the data deviates from the mean).

• Standard means typical movement from the average.

• Deviation is the distance from the mean.

• SS (sum of squares) is the sum of the squared deviations.

• Variance is the average of the squared deviations.

• Formulas:

  • For a sample:

    • $s = \sqrt{\frac{\sum(X - M)^2}{n-1}}$

    • Where:

      • $s$ = sample standard deviation

      • $X$ = each value in the sample

      • $M$ = mean of the sample

      • $n$ = number of values in the sample

  • For a population:

    • $σ = \sqrt{\frac{\sum(X - \mu)^2}{N}}$

    • Where:

      • $σ$ = population standard deviation

      • $X$ = each value in the population

      • $μ$ = mean of the population

      • $N$ = number of values in the population

Big Blossom Greenhouse Example

• Measuring rose diameters in inches for the Rose Bowl parade.

  • Bush 1: 2, 3, 4, 5, 6, 8, 10, 10

  • Bush 2: 5, 5, 5, 6, 6, 6, 7, 8

Calculation for Bush 1

• Sum of raw scores = 48

• Mean = $48 / 8 = 6$

• Calculate deviations (X - Mean): -4, -3, -2, -1, 0, 2, 4, 4

• Square the deviations: 16, 9, 4, 1, 0, 4, 16, 16

• Sum of squares (SS) = 74

• Variance = $74 / (8 - 1) = 74 / 7 = 10.57$

• Standard deviation = $\sqrt{10.57} = 3.25$

• On average, the rose size is 6 inches, with a typical deviation of about 3.25 inches.

Coefficient of Variation

• Measures relative variation (standard deviation divided by the mean).

• Useful for comparing one data series to another.

Formulas

• Sample: $(s / mean) * 100$

• Population: $(σ / μ) * 100$

Example

• New cars: mean price = $20,000, standard deviation = $6,000. Coefficient of variation = $(6000 / 20000) * 100 = 30$

• Used cars: mean price = $5,485, standard deviation = $2,730. Coefficient of variation = $(2730 / 5485) * 100 = 49$

• The smaller variation in the price of new cars (30%) indicates more stability compared to used cars (49%).