## The Z-score

The Z-Score

The Z-score also referred to as standardized raw scores is a useful statistic because not only permits to compute the probability (chances or likelihood) of raw score (occurring within normal distribution) but also it helps to compare two raw scores from different normal distributions. The Z-score is a dimensionless measure since it is derived by subtracting the population mean from an individual raw score and then this difference is divided by the population standard deviation. This computational procedure is called standardizing raw score, which is often used in the Z-test of testing of hypothesis.

Any raw score can be converted to a Z-score by

$Z Score=\frac{raw score - mean}{\sigma}$

Example 1:

If the mean = 100 and standard deviation = 10, what would be the Z-score of the following raw score

 Raw Score Z-Score 90 $\frac{90-100}{10}=-1$ 110 $\frac{110-100}{10}=1$ 70 $\frac{70-100}{10}=-3$ 100 $\frac{100-100}{10}=0$

Note that:

• If Z-score have a zero value then it means that raw score is equal to the population mean.
• If Z-score has positive value then it means that raw score is above the population mean.
• If Z-score has negative value then it means that raw score is below the population mean.

Example 2:

Suppose you got 80 marks in Exam of a class and 70 marks in another exam of that class. You are interested in finding that in which exam you have performed better. Also suppose that the mean and standard deviation of exam1 are 90 and 10 and in exam2 60 and 5 respectively. Converting both exam marks (raw scores) into standard score (Z-Score), we get

$Z_1=\frac{80-90}{10} = -1$

The Z-score results ($Z_1=-1$) shows that 80 marks are one standard deviation below the class mean.

$Z_2=\frac{70-60}{5}=2$

The Z-score results ($Z_2=2$) shows that 70 marks are two standard deviation above the mean.

From Z1 and Z2 means that in second exam student performed well as compared to the first exam. Another way to interpret the Z-score of $-1$ is that about 34.13% of the students got marks below the class average. Similarly the Z-score of 2 implies that 47.42% of the students got marks above the class average.

## Levels of Measurement (Scale of Measure)

Level of measurement (scale of measure) have been classified into four categories. It is important to understand these level of measurement, since these level of measurement play important part in determining the arithmetic and different possible statistical tests that are carried on the data. The scale of measure is a classification that describes the nature of information within the number assigned to variable.  In simple words, the level of measurement determines how data should be summarized and presented. It also indicate the type of statistical analysis that can be performed. The four level of measurement are described below:

1) Nominal Level of Measurement (Nominal Scale)

In nominal level of measurement, the numbers are used to classify the data (unordered group) into mutually exclusive categories. In other words, for nominal level of measurement, observations of a qualitative variable are measured and  recorded as labels or names.

2) Ordinal Level of Measurement (Ordinal Scale)

In ordinal level of measurement, the numbers are used to classify the data (ordered group) into mutually exclusive categories. However, it does not allow for relative degree of difference between them. In other words, for ordinal  level of measurement, observations of a qualitative variable are either ranked or rated on a relative scale and  recorded as labels or names.

3) Interval Level of Measurement (Interval Scale)

For data recorded at the interval level of measurement, the interval or the distance between values is meaningful. The interval scale is based on a scale with a known unit of measurement.

4) Ratio Level of Measurement (Ratio Scale)

Data recorded at the ratio level of measurement are based on a scale with a know unit of measurement and a meaningful interpretation of zero on the scale. Almost all quantitative variables are recorded on the ratio level of measurement.

Examples of level of measurements

Examples of Nominal Level of Measurement

• Religion (Muslim, Hindu, Christian, Buddhist)
• Race (Hispanic, African, Asian)
• Language (Urdu, English, French, Punjabi, Arabic)
• Gender (Male, Female)
• Marital Status (Married, Single, Divorced)
• Number plates on Cars/ Models of Cars (Toyota, Mehran)
• Parts of Speech (Noun, Verb, Article, Pronoun)

Examples of Ordinal Level of Measurement

• Rankings (1st, 2nd, 3rd)
• Marks Grades (A, B, C, D)
• Evaluation such as High, Medium, Low
• Educational level (Elementary School, High School, College, University)
• Movie Ratings (1 star, 2 stars, 3 stars, 4 stars, 5 stars)
• Pain Ratings (more, less, no)
• Cancer Stages (Stage 1, Stage 2, Stage 3)
• Hypertension Categories (Mild, Moderate, Severe)

Examples of Interval Level of Measurement

• Temperature with Celsius scale/ Fahrenheit scale
• Level of happiness rated from 1 to 10
• Education (in years)
• Standardized tests of psychological, sociological and educational discipline use interval scales.
• SAT scores

Examples of Ratio Level of Measurement

• Height
• Weight
• Age
• Length
• Volume
• Number of home computers
• Salary

For further details visit: Level of measurements

## Mean: Measure of Central Tendency

The measure of Central Tendency Mean (also know as average or arithmetic mean) is used to describe the data set as a single number (value) which represents the middle (center) of the data, that is average measure (performance, behaviour, etc) of data. This measure of central tendency is also known as measure of central location or measure of center.

Mathematically mean can be defined as the sum of the all values in a given dataset divided by the number of observations in that data set under consideration. The mean is also called arithmetic mean or simply average.

Example: Consider the following data set consists of marks of 15 student in certain examination.

50, 55, 65, 43, 78, 20, 100, 5, 90, 23, 40, 56, 70, 88, 30

The mean of above data values is computed by adding all these values (50 + 55 + 65 + 43 + 78 + 20 + 100 + 5 + 90 + 23 + 40 + 56 + 70 + 88 + 30 = 813) and then dividing by the number of observations added (15) which equals 54.2 marks, that is

$\frac{50 + 55 + 65 + 43 + 78 + 20 + 100 + 5 + 90 + 23 + 40 + 56 + 70 + 88 + 30 }{15}=\frac{813}{15}=54.2$

The above procedure of calculating the mean can be represented mathematically

$\mu= \frac{\sum_{i=1}^n X_i}{N}$

The Greek symbol $\mu$ (pronounced “mu”) is the representation of population mean in statistics and $N$ is the number of observations in the population data set.

The above formula is known as population mean as it is computed for whole population. The sample mean can also be computed in same manner as population mean is computed. Only the difference is in representation of the formula, that is,

$\overline{X}= \frac{\sum_{i=1}^n X_i}{n}$.

The $\overline{X}$ is representation of sample mean and $n$ shows number of observations in the sample.

The mean is used for numeric data only. Statistically the data type for calculating mean should be Quantitative (variables should be measured on either ratio or interval scale), therefore, the numbers in data set can be continuous and/ or discrete in nature.

Note that mean should not be computed for alphabetic or categorical data (data should not belong to nominal or ordinal scale). Mean is influenced by very extreme values in data, i.e. very large or very small values in data changes the mean drastically.

For other measures of central tendencies visit: Measures of Central Tencencies