CHAPTER 3: STATISTICAL CONCEPTS AND MARKET RETURNS

1.0 INTRODUCTION

The focus of the book is on 4 aspects of returns which is stated below:

1. Where the returns are centred (central tendency)
2. How does those returns vary from the centred (Dispersion)
3. Whether the distribution of returns is symmetrically shaped or lopsided (Skewness)

2.0 SOME FUNDAMENTAL CONCEPT

• Statistics is classify into descriptive and inference.
• Descriptive statistics deals with more of summarising large detail of data, and making conclusion on it
• Inference statistics deal with more of making inferences from a small units to the large units. it deal with more of probability.

2.2. Populations and Samples

• A population is defined as all members of a specified group.
• Any descriptive measure of a population characteristic is called a parameter.The investment do have alot of characteristics (parameters), but investment analysts are only interested in studying only a few, such as the mean value, the range of investment and the variance.
• A sample is a subset of a population. Just as a parameter is a descriptive measure of a population characteristic, a sample statistic (statistic, for short) is a descriptive measure of a sample characteristic.
• A sample statistic (or statistic) is a quantity computed from or used to describe a sample.

2.3. Measurement Scales

To choose the right statistical methods for summarising and analysing data, we need to we need to distinguish among different measurement scales or levels of measurement. All data measurements are taken on one of four major scales: nominal, ordinal, interval, or ratio.

• Nominal scales represent the weakest level of measurement: They categorize data but do not rank them.
• Ordinal scales sort data into categories that are ordered with respect to some characteristic. However, it does not tell us how better rank 2 is far from rank 3.
• Interval scales provide not only ranking but also assurance that the differences between scale values are equal. As a result, scale values can be added and subtracted meaningfully.
• Ratio scales represent the strongest level of measurement. They have all the characteristics of interval measurement scales as well as a true zero point as the origin.

3.0 SUMMARISING DATA USING FREQUENCY DISTRIBUTIONS

• Definition of Frequency Distribution. A frequency distribution is a tabular display of data summarized into a relatively small number of intervals.
• frequency distribution is one of the tools used in analysis of data, and it works will all types of measurement scales.
• An interval is a set of values within which an observation falls. Each observation falls into only one interval, and the total number of intervals covers all the values represented in the data.
• The actual number of observations in a given interval is called the absolute frequency, or simply the frequency.
• After the establishment of the frequency distribution, another useful way of presenting data are: the relative frequency, the cumulative frequency (also called the cumulative absolute frequency), and the cumulative relative frequency.

Relative Frequency

Relative frequency is the proportion of times a particular value or category appears in a data set, compared to the total number of values. It gives you an idea of how common a specific value is within the data set.

Example: If you surveyed 100 people about their favorite fruit and 20 said apples, the relative frequency of people who like apples is 20100=0.2010020=0.20 or 20%.

Cumulative Frequency (Cumulative Absolute Frequency)

Cumulative frequency is the running total of frequencies as you move through the data set in a particular order (usually ascending order). It shows how many observations fall below or at a certain value.

Example: Let's say we have the number of people liking different fruits:

• Apples: 20
• Bananas: 30
• Cherries: 15
• Dates: 10

The cumulative frequency is calculated as follows:

• Apples: 20 (just apples)
• Bananas: 20 (apples) + 30 (bananas) = 50
• Cherries: 50 (apples and bananas) + 15 (cherries) = 65
• Dates: 65 (apples, bananas, and cherries) + 10 (dates) = 75

So, 75 people like apples, bananas, cherries, or dates combined.

Cumulative Relative Frequency

Cumulative relative frequency is similar to cumulative frequency, but it gives the proportion (or percentage) of observations that fall below or at a certain value, compared to the total number of observations.

Example: Using the same data,

• Apples: 10020=0.20 or 20%

20100=0.20

• Bananas: 10050=0.50 or 50%

50100=0.50

• Cherries: 10065=0.65 or 65%

65100=0.65

• Dates: 10075=0.75 or 75%

75100=0.75

This tells us that 75% of the people surveyed like apples, bananas, cherries, or dates.

Summary

• Relative Frequency: How often a specific value appears compared to the total number of values.
• Cumulative Frequency: The running total of frequencies up to a certain value.
• Cumulative Relative Frequency: The running total of proportions (or percentages) up to a certain value.

These concepts help in understanding the distribution and spread of data in a dataset.

4.0 THE GRAPHIC PRESENTATION OF DATA

Part of the available tools for visualising data for quick insight is the histogram, the frequency polygon, and the cumulative frequency distribution as methods for displaying data graphically.

• Definition of Histogram. A histogram is a bar chart of data that have been grouped into a frequency distribution.

Nothing much to learn in this section…

5. MEASURES OF CENTRAL TENDENCY

As noted that frequency distributions and histograms provide a quick insight to the distribution of the data, this is a beginning method to anlayse, for a proper understanding of the distribution of the data array, we might need to calculate how the data are distributed around the a common center of the data point.

A measure of central tendency specifies where the data are centered. Measures of central tendency are probably more widely used than any other statistical measure because they can be computed and applied easily. Measures of location include not only measures of central tendency but other measures that illustrate the location or distribution of data.

The following are the common center tendency tools

1. Mean
2. Median
3. Mode
4. Weighted mean
5. Geometric mean

We also explain other useful measures of location, including quartiles, quintiles, deciles, and percentiles