LOS A and B require us to:
a. distinguish between descriptive statistics and inferential statistics, between a population and a sample, and among the types of measurement scales
b. define a parameter, a sample statistic, and a frequency distribution
1. Nature of Statistics
a. The term statistics can have two broad meanings, one referring to data and the other to the method.
b. Statistical methods include descriptive statistics and statistical inference (inferential statistics).
c. Descriptive statistics is the study of how data can be summarized effectively to describe the important aspects of large data sets. By consolidating a mass of numerical details, descriptive statistics turn data into information.
d. Statistical inference involves making forecasts, estimates, or judgments about a larger group from the smaller group actually observed.
It uses some kind of scientific procedure to draw out the samples that are representative of the population. The procedure generally includes:
i. Defining population and identifying parameters of interest
ii. Drawing sample from the population
iii. Determining the corresponding statistics and using them to estimate the parameters of the population
2. Data
a. The base of statistics is the data. The statistics basically deals with collecting and analyzing the numerical data with the purpose of deducing the inferences out of the same.
b. The data could be of two types the population data or the sample data.
c. The population data includes all the members of a specified group. For example, all the students appearing for the CFA level 1 exam is the population, as it contains all the data for the group; also, the data of this group is finite and known.
For population data, all the descriptive measures used, such as, mean, median, mode, etc. are called the population parameters and are informative in nature.
d. The sample data, on the other hand, is the subset of the population. The sample data is selected so as to represent the population data.
All the descriptive measures used for the sample data are called sample statistics, and their main purpose is to make inferences and forecasts about the population data.
3. Parameter & Statistics
3.1. Parameter
a. A parameter is a numerical quantity measuring some aspect of the population of scores.
b. The parameters are generally estimated in samples using statistical tools, are rarely known values.
c. There are many parameters for a population, but analysts are mainly concerned with some important ones such as mean, median, standard deviation, etc.
3.2. Statistic
a. A statistic is a single measure of some attribute of a sample. It is a numeric quantity calculated in a sample.
b. Statistics have two interpretations:
i. It refers to some numeric data or figures such as EBIT or EPS.
ii. It also refers to the process of collecting, organizing, presenting, analyzing, and interpreting numeric data for the purpose of making decisions.
4. Measurement Scale
There are four different types of measurement scales, i.e. nominal, ordinal, interval or ratio. These are discussed as follows:
a. Nominal: Nominal is the weakest level of measurement. It is categorical in nature. For example, different wealth group of people might be categorized differently and may be assigned different integers, such as:
High-Income Groups: 1
Mid-Income Group: 2
Lower Income Groups: 3
There is basically no correlation between each of such categories.
b. Ordinal: The ordinal scales are slightly stronger scales. It ranks or orders the categories in accordance with some characteristics, either qualitative or quantitative.
The categories may be ranked from the highest to the lowest category; such as the investments may be ranked by the analysts on the scale of 1 to 10 based on their risk-return characteristics.
This scale does not say much about the distance between the two rank categories.
c. Interval: This is a stronger scale, in the sense that it does not only provide the ranking but also the assurance that the difference between the scale values is equal. This scale assumes equal physical or psychometric distance between the intervals. For example, the cities may be categorized on the basis of their temperatures, and each category may have a distance of exactly 4 degrees.
d. Ratio: This is the strongest level of the measurement interval, and these also have the true zero point as their origin. This scale assists the application of the maximum number of statistical tools, as with this scale we can compute all the meaningful ratios and also add or subtract the amounts within the category.