Descriptive Statistics

Abstract

Statistics is an art of learning from data. One of the tasks to be performed after collecting data from any observed situation, phenomena, or interested variable is to analyze that data to extract some useful information. Statistical analysis is one of the most applied tools in the industry, decision making, planning, public policy, etc. Many practical applications start from analyzing data, which is the main information source. Given this data, the analyst should be able to use this data to have an idea of what the collected data have to say, either by providing a report of his/her findings or making decisions.

Problems

3.1

Marks (out of 100) obtained by a class of students in course “Probability and Statistics” is given in Table 3.13.

• 1. Write the frequency table for interval of ten marks, i.e., 1–10, 11–20, and so on.

• 2. Draw the histogram of the distribution.

• 3. Comment on the symmetry and peakedness of the distribution after calculating appropriate measures.

Table 3.13 Data for Problem 3.1

3.2

Let X equal the duration (in min) of a telephone call that is received between midnight and noon and reported. The following times were reported

$$\begin{aligned} 3.2,~0.4,~1.8,~0.2,~2.8,~1.9,~2.7,~0.8,~1.1,~0.5,~1.9,~2,~0.5,~2.8,~1.2,~1.5,~0.7,~1.5,~2.8,~1.2 \end{aligned}$$

Draw a probability histogram for the exponential distribution and a relative frequency histogram of the data on the same graph.

3.3

Let X equal the number of chips in a chocolate chip cookies. Hundred observations of X yielded the following frequencies for the possible outcome of X.

$$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline Outcome (x) &{}0&{}1&{}2&{}3&{}4&{}5&{}6&{}7&{}8&{}9 &{}10 \\ \hline Frequency &{}0&{}4&{}8&{}15&{}16&{}19&{}15&{}13&{}7&{}2&{}1 \\ \hline \end{array} $$

• 1. Use these data to graph the relative frequency histogram and the Poisson probability histogram.

• 2. Do these data seem to be observations of a Poisson random variable with mean $\lambda$. Find $\lambda$.

3.4

Show that the total sum of the distances between each data and its mean, namely d, $d_i=x_i-\overline{x}$, is zero.

3.5

Calculate the value of commonly used statistics to find measure of spread for the runs scored by the Indian Cricket Team based on their scores in last 15 One Day Internationals while batting first. The data are shown in Table 3.14.

Table 3.14 Data for Problem 3.5

3.6

A mobile phone company examines the ages of 150 customers to start special plans for them. Consider frequency table shown in Table 3.15.

• 1. Draw the histogram for the data.

• 2. Estimate the mean age for these policyholders.

• 3. Estimate the median age for these policyholders.

Table 3.15 Frequency table for Problem 3.6

3.7

A sample of 10 claims in an insurance company had mean and variance of 5,478 and 1,723, respectively. On reconciliation, it was found that one claim of 3,250 was wrongly written as 4,250. Calculate the mean and standard deviation of the sample with correct values.

3.8

Suppose a state government wants to analyze the number of children in families for improving their immunization program. They analyze a group of 200 families and report their findings in the form of a frequency distribution shown in Table 3.16

• 1. Draw the bar chart for the following data and calculate the total number of children.

• 2. Calculate mean, mode, and median of the data.

• 3. Calculate coefficient of kurtosis and coefficient of skewness in the above data.

Table 3.16 Frequency table for Problem 3.8

3.9

An insurance company wants to analyze the claims for damage due to fire on its household content’s policies. The values for a sample of 50 claims in Rupees are shown in Table 3.17.

Table 3.17 Data for Problem 3.9

Table 3.18 displays the grouped frequency distribution for the considered data.

Table 3.18 Grouped frequency distribution of the data given in Table 3.17

• 1. What is the range of the above data?

• 2. Draw a bar graph for Table 3.18.

• 3. For the data given in Table 3.18 if instead of equal-sized groups, we had a single group for all value below 250, how would this bar be represented?

• 4. Calculate the mean, median, mode, and sample geometric mean.

• 5. Calculate the sample standard deviation and sample variance.

• 6. Calculate the coefficient of variation.

3.10

Suppose the Dean of a college wants to know some basic information about the height of the students of the last six year groups. To do so, twenty students were selected from each group and their heights were measured, so a total of 120 observations divided into six groups are taken. The obtained results are presented in Table 3.19.

The Dean needs to know if there is any evidence that some students have the required height to play sports such as basketball, volleyball, and swimming. On the other hand, he is looking for some useful information to plan some requirements such as uniforms, shoes and caps. Provide a descriptive report of the information found in the provided data. If possible, also provide graphics and recommendations.

Table 3.19 Data for Problem 3.10

3.11

Show that the variance of any variable $s^2$ can be expressed as

$$\begin{aligned} s^2=\dfrac{\displaystyle {\sum _{i=1}^{n}x_i^2}-\dfrac{\left( \displaystyle {\sum _{i=1}^{n}x_i} \right) ^2}{n}}{n-1} . \end{aligned}$$

Test out this result against all data provided in Problem 3.10.

3.12

The department of analysis of a taxicab company has the records of 15 drivers, which are shown in Table 3.20. Those records include the following information: distance (Km), amount of crashes, amount of fines received per driver, amount of visits to garage for repairs (V.T.G), and days of operation (D.O.O).

Table 3.20 Recorded data for Problem 3.12

Provide a statistical report of the information provided by data. The department of analysis of the company is looking for information useful to identify good drivers and the efficiency of the fleet.

3.13

Consider Problem 3.12. The department of analysis of the company suggests to compare all variables to each others using graphical analysis. The idea is to find some patterns whose information should be useful to identify possible relationships among variables. Provide only interesting graphics alongside with a description of their findings.