The post is about the MCQs Basic Statistics. There are 20 multiple-choice questions with answers covering the topics of coefficient of variation, ungrouped and grouped data, distribution of data, primary and secondary data, data collection methods, variables, reports, sample and population, surveys, and data collection methods. Let us start with MCQs Basic Statistics quiz.
Online MCQs about Basic Statistics with Answers
Online MCQs Basic Statistics with Answers
The mean of a distribution is 14 and the standard deviation is 5. What is the value of the coefficient of variation?
The mean of a distribution is 23, the median is 24, and the mode is 25.5. It is most likely that this distribution is:
According to the empirical rule, approximately what percent of the data should lie within $\mu \pm 2\sigma$?
If a distribution is abnormally tall and peaked, then it can be said that the distribution is:
The sum of dots, when two dice are rolled, is
The number of accidents in a city during 2010 is
The first-hand and unorganized form of data is called
The data which have already been collected by someone are called
Census reports used as a source of data is
The grouped data is also called
Primary data and ——— data are the same
The questionnaire survey method is used to collect
Data collected by NADRA to issue computerized identity cards (CICs) are
A parameter is a measure which is computed from
Given $X_1=12,X_2=19,X_3=10,X_4=7$, then $\sum_{i=1}^4 X_i$ equals?
A chance variation in an observational process is
A constant variable can take values
A specific characteristic of a population is called
The listing of the data in order of numerical magnitude is called
A variable that assumes any value within a range is called
In this post, we will discuss about properties of Arithmetic mean with Examples.
Arithmetic Mean
The arithmetic mean, often simply referred to as the mean, is a statistical measure that represents the central value of a dataset. The arithmetic mean is calculated by summing all the values in the dataset and then dividing by the total number of observations in the data.
Table of Contents
The Sum of Deviations From the Mean is Zero
Property 1: The sum of deviations taken from the mean is always equal to zero. Mathematically $\sum\limits_{i=1}^n (x_i-\overline{x}) = 0$
Consider the ungrouped data case.
Obs. No.
$X$
$X_i-\overline{X}$
1
81
-19
2
100
0
3
96
-4
4
108
8
5
90
-10
6
102
2
7
104
4
8
103
3
9
100
0
10
109
9
11
91
-9
12
116
16
Total
$\sum X_i = 1200$
$\sum\limits_{i=1}^n (X_i-\overline{X})=0$
For grouped data $\overline{X} = \sum\limits_{i=1}^k f_i(X_i -\overline{X}) =0$, where for grouped data $\overline{X} =\frac{\sum\limits_{i=1}^n M_i f_i}{\sum\limits_{i=1}^n f_i}$. Suppose, we have the following grouped data
For combined mean, not all the data set needs to be ungrouped or grouped. It may be possible that some data sets are ungrouped and some data sets are grouped.
Sum Squared Deviations from the Mean are Always Minimum
Property 3: The sum of the squared deviations of the observations from the arithmetic mean is minimum, which is less than the sum of the squared deviations of the observations from any other values. In other words, the sum of squared deviations from the mean is less than the sum of squared deviations from any other value. Mathematically,
where $A$ is any arbitrary value, also known as provisional mean. For this condition, $A$ is not equal to the arithmetic mean.
Note that the difference between the sum of deviations and the sum of squared deviations is that in the sum of deviations we take the difference (subtract) of each observation from the mean and then sum all the differences. In the sum of squared deviations, we take the difference of each observation from the mean, then take the square of all the differences, and then sum all the resultant values at the end.
From the above calculations, it can observed that $\Sigma (X_i – \overline{X})^2 < \Sigma (X_i – 90)^2 < \Sigma (X_i – 99)^2$.
No Resistant to Outliers
Property 4: The arithmetic mean is not resistant to outliers. It means that the arithmetic mean can be misleading if there are extreme values in the data.
Arithmetic Mean is Sensitive to Outliers
Property 5: The arithmetic mean is sensitive to extreme values (outliers) in the data. If there are a few very large or very small values, they can significantly influence the mean.
The Affect of Change in Scale and Origin
Property 6: If a constant value is added or subtracted from each data point, the mean will be changed by the same amount. Similarly, if a constant value is multiplied or divided by each data point, the mean will be multiplied or divided by the same constant.
Unique Value
Property 7: For any given dataset, there is only one unique arithmetic mean.
In summary, the arithmetic mean is a widely used statistical measure (a measure of central tendency) that provides a central value for a dataset. However, it is important to be aware of the properties of arithmetic mean and its limitations, especially when dealing with data containing outliers.
FAQs about Arithmetic Mean Properties
Explain how the sum of deviation from the mean is zero.
What is meant by unique arithmetic mean for a data set?
What is arithmetic mean?
How combined mean of different data sets can be computed, explain.
Elaborate Sum of Squared Deviation is minimum?
What is the impact of outliers on arithmetic mean?
How does a change of scale and origin change the arithmetic mean?
A one factor design (also known as a one-way ANOVA) is a statistical method used to determine if there are significant differences between the means of multiple groups. In this design, there is one independent variable (factor) with multiple levels or categories.
Table of Contents
Suppose $y_{ij}$ is the response is the $i$th treatment for the $j$th experimental unit, where $i=1,2,\cdots, I$. The statistical model for a completely randomized one-factor design that leads to a One-Way ANOVA is
$$y_{ij} = \mu_i + e_{ij}$$
where $\mu_i$ is the unknown (population) mean for all potential responses to the $i$th treatment, and $e_{ij}$ is the error (deviation of the response from population mean).
The responses within and across treatments are assumed to be independent and normally distributed random variables with constant variance.
One Factor Design’s Statistical Model
Let $\mu = \frac{1}{I} \sum \limits_{i} \mu_i$ be the grand mean or average of the population means. Let $\alpha_i=\mu_i-\mu$ be the $i$th group treatment effect. The treatment effects are constrained to add to zero ($\alpha_1+\alpha_2+\cdots+\alpha_I=0$) and measure the difference between the treatment population means and the grand mean.
Therefore the one way ANOVA model is $$y{ij} = \mu + \alpha_i + e_{ij}$$
From this model, the hypothesis of interest is whether the population means are equal:
$$H_0:\mu_1=\mu_2= \cdots = \mu_I$$
The hypothesis is equivalent to $H_0:\alpha_1 = \alpha_2 =\cdots = \alpha_I=0$. If $H_0$ is true, then the one-way ANOVA model is
$$ y_{ij} = \mu + e_{ij}$$ where $\mu$ is the common population mean.
One Factor Design Example
Let’s say you want to compare the average test scores of students from three different teaching methods (Method $A$, Method $B$, and Method $C$).
Independent variable: Teaching method (with three levels: $A, B, C$)
Dependent variable: Test scores
When to Use a One Factor Design
Comparing means of multiple groups: When one wants to determine if there are significant differences in the mean of a dependent variable across different groups or levels of a factor.
Exploring the effect of a categorical variable: When one wants to investigate how a categorical variable influences a continuous outcome.
Assumptions of One-Factor ANOVA
Normality: The data within each group should be normally distributed.
Homogeneity of variance (Equality of Variances): The variances of the populations from which the samples are drawn should be equal.
Independence: The observations within each group should be independent of each other.
When to Use One Factor Design
When one wants to compare the means of multiple groups.
When the independent variable has at least three levels.
When the dependent variable is continuous (e.g., numerical).
Note that
If The Null hypothesis is rejected, one can perform post-hoc tests (for example, Tukey’s HSD, Bonferroni) to determine which specific groups differ significantly from each other.
Remember: While one-factor designs are useful for comparing multiple groups, they cannot establish causation.
