Untitled

STTN111 Chapter 4: Descriptive Measures of Location

1. Overview of Concepts

• Objective: After completing this unit, you should be able to:
  • Describe measures of location.
  • Identify and describe different measures of location.
  • Explain how different measures of location describe data.
  • Calculate different measures of location.

2. Chapter Structure

2.1 Sections of Chapter 4

• 4.1 Introduction
• 4.2 Arithmetic Mean
• 4.3 Mode
• 4.4 Median
• 4.5 Quantiles
• 4.6 Comparison of Measures

3. Definitions

3.1 Introduction

• Parameter: A summarizing value calculated from a whole population.
• Statistic: A summarizing value calculated from a sample.
• Measure of Location (Measure of Position): A single number that attempts to portray the center of a population or sample.

3.2 Types of Data

• Ungrouped Data: Raw data not divided into classes.
• Grouped Data: Data that has been divided into classes.

4. Arithmetic Mean

4.1 Definition

• Arithmetic Mean ($\bar{x}$): The arithmetic mean of n values ($x_1, x_2, ext{…}, x_n$) is calculated as:
  $\bar{x} = \frac{x_1 + x_2 + ext{…} + x_n}{n}$

4.2 Calculation Method

4.2.1 Ungrouped Data

• Calculated easily using the formula.
• Example: The distances of 8 javelin throwers:
  • Distances: 71.3, 68.4, 78.8, 58.4, 66.1, 84.7, 74.4, 75.9
  • Mean calculation:
    $\bar{x} = \frac{71.3 + 68.4 + 78.8 + 58.4 + 66.1 + 84.7 + 74.4 + 75.9}{8} = 72.24$

4.3 Grouped Data

4.3.1 Continuous Data

• Steps to calculate mean:
  1. Determine the class midpoint for each class (m).
  2. Calculate the sum of the observations in each class:
     $f_i m_i$
  3. Add totals of classes together to get the estimated sum of all n observations.
  4. Divide total by n for the mean.
• The formula for arithmetic mean for grouped continuous data is:
  ar{x} = rac{egin{pmatrix}f_1 m_1 + f_2 m_2 + ext{…} + f_k m_k ext{ } ext{where } n = ext{ total number of observations} ext{ } ext{and } i ext{ } represents class } ext{ and frequency associated}{ ext {with class}} ext{ } k

4.4 Example Calculation

• Example: Monthly electricity expenditure of 65 families:
  • Class intervals and frequencies:
    | Interval | Midpoint (m) | Frequency (f) | f * m |
    |-------------|---------------|----------------|--------|
    | [250, 350) | 300 | 5 | 1500 |
    | [350, 450) | 400 | 10 | 4000 |
    | [450, 550) | 500 | 16 | 8000 |
    | [550, 650) | 600 | 13 | 7800 |
    | [650, 750) | 700 | 10 | 7000 |
    | [750, 850) | 800 | 7 | 5600 |
    | [850, 950) | 900 | 4 | 3600 |
  • Total frequency: 65
  • Total sum of observations: 37500
  • Mean calculation:
    $\bar{x} = \frac{37500}{65} = 576.92$

4.5 Properties of Mean

• All observations are utilized.
• Sensitive to outliers.
• Unique for each data set.
• Exists for all quantitative data.
• Examples showing sensitivity to outliers:
  • For 2, 3, 4, 5, mean = 3.5;
  • For 2, 3, 4, 50, mean = 14.75.

5. Mode

5.1 Definition

• Mode: The value with the highest incidence in a dataset.
• Example with data and frequency demonstrating mode calculation.

5.2 Ungrouped Data

• More common for discrete data.
• Example of number of bedrooms in houses demonstrating mode.

5.3 Grouped Data

5.3.1 Continuous

• Calculate mode from frequency data indicating modal interval.
• Example given with frequencies.

5.4 Properties of Mode

• Not influenced by outliers.
• Could have multiple modes in some datasets.
• Generally used with discrete data (ordinal and nominal scales).

6. Median

6.1 Definition

• Median ($x$): The middle value when observations are arranged in order.
• If n is odd: the middle observation.
• If n is even: average of two middle observations.

6.2 Calculation for Ungrouped Data

• Example showing how to find median with odd and even sets.

6.3 Grouped Data

• Determining the median through cumulative frequency polygons.
• Steps for determining value of median from cumulative frequencies.

7. Quantiles

7.1 Definition

• Quantiles divide data into equal parts:
  • Median: 2 parts
  • Quartiles: 4 parts
  • Deciles: 10 parts
  • Percentiles: 100 parts

7.2 Calculation for Ungrouped Data

• Formulae for calculating different quartiles and their positions based on dataset size.

7.3 Calculation for Grouped Data

• Determine values using cumulative frequency polygon.
• Example illustrating the calculation using cumulative frequency data from households.

8. Comparison of Measures

8.1 Distribution Characteristics

• Symmetrical Distribution: Mean, median, and mode are equal.
• Skewed Distribution:
  • Positively Skewed: Mean > median > mode
  • Negatively Skewed: Mean < median < mode
  • Practical implications when analyzing data leads to the use of different measures based on skewness.