## Calculating Mean, Median, Mode, Quartiles, and More

Question 1:

Suppose that the data for analysis includes the attribute age. The age values for the data tuples are (in increasing order) 13, 15, 16, 16, 19, 20, 20, 21, 22, 22, 25, 25, 25, 25, 30, 33, 33, 35, 35, 35, 35, 36, 40, 45, 46, 52, 70.

(a)What is the mean of the data? What is the median?

(b)What is the mode of the data? Comment on the data’s modality (i.e., bimodal, trimodal, etc.).

(c)What is the midrange of the data?

(d)Can you find (roughly) the first quartile (Q1) and the third quartile (Q3) of the data?

(e)Give the five-number summary of the data. (f) Show a boxplot of the data.

(g) How is a quantile-quantile plot different from a quantile plot ?

Question 2: In many applications, new data sets are incrementally added to the existing large data sets. Thus an important consideration for computing descriptive data summary is whether a measure can be computed efficiently in incremental manner. Use count, standard deviation, and median as examples to show that a distributive or algebraic measure facilitates efficient incremental computation, whereas a holistic measure does not.

Question 3: In real-world data, tuples with missing values for some attributes are a common occurrence. Describe various methods for handling this problem.

Question 4: The age values for the data tuples are (in increasing order) 13, 15, 16, 16, 19, 20, 20, 21, 22, 22, 25, 25, 25, 25, 30, 33, 33, 35, 35, 35, 35, 36, 40, 45, 46, 52, 70. Using this data for age, answer the following:

(a)Use smoothing by bin means to smooth the above data, using a bin depth of 3. Illustrate your steps. Comment on the effect of this technique for the given data.

(b)How might you determine outliers in the data?

(c)What other methods are there for data smoothing ?

Question 4: Discuss issues to consider during data integration.

Question 5: Suppose a hospital tested the age and body fat data for 18 randomly selected adults with the following result

age 23 23 27 27 39 41 47 49 50

%fat 9.5 26.5 7.8 17.8 31.4 25.9 27.4 27.2 31.2

age 52 54 54 56 57 58 58 60 61

%fat 34.6 42.5 28.8 33.4 30.2 34.1 32.9 41.2 35.7

(a)Calculate the mean, median and standard deviation of age and %fat.

(b)Draw the boxplots for age and %fat.

(c)Draw a scatter plot and a q-q plot based on these two variables.

(d)Normalize the two variables based on z-score normalization.

(e)Calculate the correlation coefficient (Person’s product moment coefficient). Are these two variables positively or negatively correlated?

Question 6: State why, for the integration of multiple heterogeneous information sources, many companies in industry prefer the update-driven approach (which constructs and uses data warehouses), rather than the query-driven approach (which applies wrappers and integrators). Describe situations where the query-driven approach is preferable over the update-driven approach.

Question 7: Briefly compare the following concepts. You may use an example to explain your point(s).

(a)Snowflake schema, fact constellation, starnet query model

(b)Data cleaning, data transformation, refresh

(c)Enterprise warehouse, data mart, virtual warehouse

Question 8: Suppose that a data warehouse consists of the three dimensions time, doctor, and patient, and the two measures count and charge, where charge is the fee that a doctor charges a patient for a visit.

(a)Enumerate three classes of schemas that are popularly used for modeling data warehouses.

(b)Draw a schema diagram for the above data warehouse using one of the schema classes listed in (a).

a)Users are mainly interested in four particular dimensions, each having three frequently accessed levels for rolling up and drilling down. How would you design a data cube structure to efficiently support this preference?

b)At times, a user may want to drill through the cube, down to the raw data for one or two particular dimensions. How would you support this feature?

Question 13: What are the differences between the three main types of data warehouse usage: information processing, analytical processing, and data mining? Discuss the motivation behind OLAP mining (OLAM).

Question 14: A data cube, C, has n dimensions, and each dimension has exactly p distinct values in the base cuboid. Assume that there are no concept hierarchies associated with the dimensions.

a)What is the maximum number of cells possible in the base cuboid?

b)What is the minimum number of cells possible in the base cuboid?

c)What is the maximum number of cells possible (including both base cells and aggregate cells) in the data cube, C?

Question 15:

Suppose that a base cuboid has three dimensions A, B, C, with the following number of cells: |A| =

1, 000, 000, |B| = 100, and |C| = 1000. Suppose that each dimension is evenly partitioned into 10 portions for chunking.

(a) Assuming each dimension has only one level, draw the complete lattice of the cube.

(b) If each cube cell stores one measure with 4 bytes, what is the total size of the computed cube if the cube is dense?

Question 16: A flight data warehouse for a travel agent consists of six dimensions: traveller, departure (city), departure time, arrival, arrival time, and flight; and two measures: count, and avg fare, where avg fare stores the concrete fare at the lowest level but average fare at other levels.

(a)Suppose the cube is fully materialized. Starting with the base cuboid [traveller, departure, departure time, arrival, arrival time, flight], what specific OLAP operations (e.g., roll-up flight to airline) should one perform.

The given dataset of attribute age is

13, 15, 16, 16, 19, 20, 20, 21, 22, 22, 25, 25, 25, 25, 30, 33, 33, 35, 35, 35, 35, 36, 40, 45, 46, 52, 70.

a)The mean of the attribute age is calculated by summing all the values and then dividing the sum by number of values. This is calculated in excel as given below.

Hence, the mean of the data as calculated in excel is

Mean = 29.96296

Median = 25

b)The mode of the data is given by,

The number which has the highest frequency in the data set.

Now, two numbers 25 and 35 are in the dataset for 4 times which is highest number of occurrence. Hence, the data is bimodal as there are two modes which are 25 and 35.

c)The midrange of the data set is given by,

Middle range = (range/2) = (maximum – minimum)/2 = 28.5.

Now, the midrange value = minimum + (range/2) = 28.5 + 13 = 41.5.

d) The first quartile is the mid-value of the minimum and median value and the third quartile is the mid-value of the median and the maximum value.

Q1 = 20 and Q3 = 35.

e) The five-number summary of data is given below.

Minimum |
13 |

Q1 |
20 |

Median |
25 |

Q3 |
35 |

Maximum |
70 |

Hence, 5 number summary is 13,20,25,35, 70.

The boxplot is shown below.

g)In the quantile plot the quantiles of a set of values are displayed. The sample quantiles are in Y axis and fraction of sample they correspond are in the x axis.

In the Quantile-quantile (Q-Q) plot the quantiles of two sets of data are compared. The comparison is more than just comparing the mean and median, the shapes of the two datasets are also compared.

The distributive measures are those measures which gives the same results when applying on the whole dataset and in the subgroups of dataset. For example, the functions like count, sum, min and max are distributive measures as measure of the whole dataset is obtained by just summing up the values for the subgroups. The Algebraic measure are the measures which can be computed for the whole dataset by using an algebraic function of n arguments (n is positive integers) in the subgroups. Average is an algebraic measure as average = sum / count. If for some measure there exists no algebraic function which can compute the measure of total dataset by using the measures in subgroups then those are the holistic measures. Some examples of the holistic measures are mode, median and rank.

## Dealing with Missing Values and Smoothing Data

Missing values in a dataset is common for real world scenarios as all the sample observations are complete in very rare cases. Now, common practise to deal with missing values is to remove those missing pairs and do the analysis with the rest values which are complete. Now, this gives good accuracy when the missing values are within 5% of the sample size. However, when the missing values are very large then the case is discarded and the data collection is performed again. In case of univariate analysis an intuitive guess can be applied if there is less number of missing values to have sufficient amount of accuracy in the analysis.

The given dataset is

13, 15, 16, 16, 19, 20, 20, 21, 22, 22, 25, 25, 25, 25, 30, 33, 33, 35, 35, 35, 35, 36, 40, 45, 46, 52, 70.

- a) Now, in the smoothing by means technique the whole dataset is divided into several parts with number of values in a part equal to the bin depth. Then the average of the values in a part is calculated and all the values in that part is replaced with the average.

Now, bin depth = 3.

Hence, smoothing by bin means:

bin 1 = 14.66666667, 14.66666667, 14.66666667

bin 2 = 18.33333333, 18.33333333, 18.33333333

bin 3 = 21, 21, 21

bin 4 = 24, 24, 24

bin 5 = 26.66666667, 26.66666667, 26.66666667

bin 6 = 33.66666667, 33.66666667, 33.66666667

bin 7 = 35, 35, 35

bin 8 = 40.33333333, 40.33333333, 40.33333333

bin 9 = 56, 56, 56.

Smoothing data by means will move the noises in the data and a smooth curve can be found for the given data.

- b)The outliers in a data set are the points which are over Q3 + 1.5*IQR and below Q1 – 1.5*IQR.

Where, Q1 = 1^{st} quartile, Q3 = 3^{rd} quartile

IQR = Q3 – Q1

- c) The other types of data smoothing are smoothing by equal frequency and smoothing by bin boundaries.

Different issues in data integration are varied information, lack of manpower, storeroom competence shortage and awful information.

Question 5:

Age |
% Fat |

23 |
9.5 |

23 |
26.5 |

27 |
7.8 |

27 |
17.8 |

39 |
31.4 |

41 |
25.9 |

47 |
27.4 |

49 |
27.2 |

50 |
31.2 |

52 |
34.6 |

54 |
42.5 |

54 |
28.8 |

56 |
33.4 |

57 |
30.2 |

58 |
34.1 |

58 |
32.9 |

60 |
41.2 |

61 |
35.7 |

- a) The mean and standard deviation of age and %fat is calculated in excel as given below.

mean age |
46.44444 |

median age |
51 |

std age |
13.21862 |

mean % fat |
28.78333 |

std %fat |
9.254395 |

median % fat |
30.7 |

b) The boxplots of age and %fat are constructed using Minitab as given below.

c) The scatter plot is constructed where the age is considered as independent variable(x axis) and the %fat is considered as dependent variable (y axis).

The q-q plot of age and % fat is constructed in MATLAB and given below.

Matlab code:

>> age = [23 23 27 27 39 41 47 49 50 52 54 54 56 57 58 58 60 61];

>> percent_fat = [9.5 26.5 7.8 17.8 31.4 25.9 27.4 27.2 31.2 34.6 42.5 28.8 33.4 30.2 34.1 32.9 41.2 35.7];

>> qqplot(age,percent_fat)

>> xlabel('age'); ylabel('% fat')

>> title('Q-Q plot')

d) The normalized z score for a dataset xi is given by the following formula.

Zi = (xi – min(x))/(max(x) – min(x))

Now, by this following formula the age variable is normalized.

norm_age |
norm_%_fat |

0 |
0.048991354 |

0 |
0.538904899 |

0.105263158 |
0 |

0.105263158 |
0.288184438 |

0.421052632 |
0.680115274 |

0.473684211 |
0.521613833 |

0.631578947 |
0.564841499 |

0.684210526 |
0.55907781 |

0.710526316 |
0.674351585 |

0.763157895 |
0.772334294 |

0.815789474 |
1 |

0.815789474 |
0.60518732 |

0.868421053 |
0.737752161 |

0.894736842 |
0.645533141 |

0.921052632 |
0.757925072 |

0.921052632 |
0.723342939 |

0.973684211 |
0.962536023 |

1 |
0.804034582 |

## Visualizing Data through Box Plots, Scatter Plots, and Q-Q Plots

e) The Pearson’s correlation coefficient is calculated by the correl function in excel.

Correl(age,%fat) = 0.817618796.

Hence, the two variables are positively correlated. This is also evident from the scatterplot of the two variables.

The two approaches for the integration of different heterogeneous sources of information are update driven approach and the query driven approach. Update driven approach is the most popular and efficient method as the information is copied, process and restructured from many heterogeneous sources before doing the data analysis. Thus no interface is required for query processing in update driven approach. Thus Update driven approach is preferred in many industries for its high performance and efficient solution. The query driven approach is used for building wrappers and the integrators on the top of the heterogeneous databases. This is basically needed when the query needs are unknown.

- a)

Star schema |
Snowflake Schema |
Fact Constellation |

There is only one central fact table and each dimension has one table. |
There is only one central fact table and each dimension has one table. |
Dimension tables are shared by multiple fact tables. |

The hierarchies are not directly captured. |
Vary form Star schema. |
Viewed as collection of stars and known as galaxy schema. |

Each of the fact points are in one tuple in a single dimension with additional attributes. |
The dimension tables are normalized and then split into additional tables. |
The schema is for sophisticated applications. |

b) Data refresh is a method where entire data solution is re-built in several large batches from scratch. The initial load of batch processing code is used for refreshing data.

Data cleaning is basically a process of detecting and then correcting corrupted data records from a table or a database and the identification of incorrect, incomplete or any irrelevant parts of data. Then these parts of data are either replaced or modified or deleted. These corrupted part of data are known as dirty or coarse data.

In data transformation process data is transformed to a form which is suitable for applying the data mining process. Some of the data transformation strategies are smoothing, aggregation, generalization, normalization, attribute construction. By smoothing noise is removed from data. In aggregation process the summary operations are applied to data. In the process of generalization the low levels of data are efficiently replaced with high level of data by the hierarchy climbing process. Normalization is a process of scaling data such that it falls under the range 0 to 1. In case of attribute reconstruction new types of attributes are made from attributes which are available.

c) The enterprise warehouse is particularly a system which is used for data analysis and reporting and it is the most fundamental component of the business. It is the central repositories of the integrated data extracted from one or more disparate source.

The data mart is basically an access pattern or a structure that is built for particularly data warehouse environments for retrieving the data that is more client faced. It is basically a subset of a large data warehouse and directed to a particular team or business line.

## Data Transformation Techniques and Processes

- a)The most popular classes of schema for data warehouse modelling are
- i) Star schema: This is made with central fact table and dimension tables. The two sets of columns in the fact table of this schema contain foreign keys which are known as the primary keys for the dimension tables and measure of facts respecotively.
- ii) Snowflake schema: This is very similar to the star schema having just two levels of the dimension tables. The tables of dimension are normalized and then split in two by using relevant data.

iii) Fact constellation: This is made using two interconnected Fact tables associated with dimension tables.

- b) Now, the star schema diagram for data house as given in the question is given below.

- c) The specific operations in order is given below.
- Rolling up from day then to month and then to year
- Slice year = “2004”
- Individual patient to all patient roll up
- Patient slice = “all”
- Obtaining the list of total collected fee by every doctor in the year 2004.
- d) SQL query assuming the data is stored in a relational database using star schema is given below.

SQL

Select doctor, Sum(charge)

From fee

Where year = 2004

Group by doctor

- a) The snowflake schema diagram for data warehousing is shown below.

- b) Starting with the base cuboid, the specific OLAP operations that must be performed for listing the average operations are given below.

[student, course, semester, instructor]

- from course_key to major roll-up on course
- from student_key to university roll-up on student.
- Student in department = “CS” and university = “Big University” dicing on course.
- from University to student name drilling down on student
- c) As each dimension has 5 level and each level there are 4 cuboids hence the cube will contain 5^4 = 625 cuboids.

- a) The star schema diagram for the given data warehouse is the following.

- b) The specific OLAP operations are given below.
- date_id to year roll up on date
- game_id to all roll up on game
- location_id to location_name roll up on location
- spectator_id to status roll up on spectator
- Location name = “GM_Place”, year = 2004 and Dice status = “students”
- c) The Bitmap indexing is very useful technique in data warehousing. This technique is beneficial in low-cardinally domains. Now, as an example when the dimension location is indexed by bitmap then joining, comparison and aggregation operations over the locations are reduced in bit arithmetic and thus the processing time is reduced.

- a) The ROLAP technique is basically consist of intermediate servers in between the back end and front end server and thus it uses a relational or extended-relational DBMS for storing and managing the warehouse data. The MOLAP implementation has servers that supports the multidimensional view of the data by the multidimensional storage engines. The HOLAP is the combination of ROLAP and MOLAP and thus large volume of detailed data can be stored in HOLAP.
- b) Using ROLAP the data warehouse generation is implemented with the use of relational and extended-relational DBMS by the use of fact tables.

- a) The particular structure of the data cube is for supporting the preference that would be for using partial level of materialization or to select computation cuboids. The total storage space can be reduced when only the proper subsets among the whole set of cuboids are calculated. This also reduces the response time.
- b) As the user’s choice is to drill within one or two dimensions, the feature can be supported with the computation of the required cuboids those are on the fly. This particular feature is not needed by the user for many times and thus the extra time required for computation of one or two dimensions those are on the fly can be easily accepted.

The information processing have the queries for finding and reporting information by the use of crosstabs, graphs, charts and tables. The technique of analytical processing use different types of OLAP operations like slice and dice, drill down, roll up and historical data pivoting for the case of analysis of multidimensional warehouse data. The data mining uses the discovery of the knowledge for finding patterns which are hidden and associations, construction of analytical models, performance of prediction and classification.

On-line selection of the data mining functions basically allows the users who do not know the different knowledge that should be mined by the flexibility for selecting the desired functions of data mining and then data mining tasks are dramatically swapped.

- a) The maximum number of cells possible in the cuboid which is regarded as base is p^n. This the largest number of distinct tuples that can be formed from p distinct numbers in every dimension.
- b) p tuples are needed for containing p distinct values in each of the dimensions. Hence, minimum number of cells possible in the base cuboid is p.
- c) Additionally, a ∗ can also be chosen other than the p distinct values for each dimension. Thus maximum number of cells possible for n dimensions is (p+1)^n.
- d) The minimum cell number is the number when there are p cells in every cuboid other than the apex that has a single cell.

- a) The complete lattice structure of the cube is given below.

- b) The total size of the computed cube is calculated below.

All = 1

A = 1,000,000, B = 100 and C = 1000. Hence, subtotal = 1,001,100.

AB = 100,000,000, BC = 100,000, AC = 1,000,000,000 and hence the subtotal = 1,100,100,000.

ABC = 100,000,000,000.

Hence, total size in bytes = 101,101,101,101 ×4 bytes = 404,404,404,404 bytes.

- c) The least amount of space is required in computation order B-C-A. The diagram is shown below.

The total space required = 100 * 1000 + 1,000,000 * 10 + 100*10000 = 20,100,000 = 80,400,000 bytes.

- a)The OLAP operations are listed in a sequence.
- Level of category and dice on “business” roll up on traveller.
- In the level of city and on dice “L.A” on departure.

iii. In the level of “ANY” roll up on the “departure_time”

- In the level of “ANY” roll up on the arrival.
- In the level of “ANY” roll-up on the arrival time
- In the level of company and dice on “AA” roll up on flight.

vii. In the level of individual drill down on traveller.

viii. In the level of month drill down on the departure-time.

- b) The two constraints are min_sup = 10 and avg_fare > 500. When using the iceberg cubing algorithm like BUC we get avg_fare > 500. Now, as avg_fare is anti-monotonic, hence, this should be converted to a top-k-avg or avg^10(fare) > 500. Now for pruning the computation of the cube binning is added to min_sup.

- a) The itemsets that are frequent by using Apriori and FP-growth are given below.

Apriori:

The FP-growth tree is shown below

The FP growth table is given below.

item |
Conditional pattern base |
Conditional tree |
Frequent pattern |

y |
{{k,e,m,o:1},{k,e,o:1},{k,m:1}} |
k:3 |
{k,y:3} |

o |
{{k,e,m:1},{k,e:2}} |
k:3,e:3 |
{k,o:3},{e,o:3},{k,e,o:3} |

m |
{{k,e:2},{k:1}} |
k:3 |
{k,m:3} |

e |
{{k:4}} |
k:4 |
{k,e:4} |

Apriori does many scans in the database, however the FP-growth can build a FP-Tree with just only one scan. Hence, FP-growth is more time efficient than Apriori. Apriori needs to generate candidates and those generations are expensive but there is no candidate generation in FP-growth technique.

- a)The rule template for granularity of item_category is

∀X ∈ transaction, buys(X,item1) ∧ buys(X,item2) => buys(X,item3) [s,c]

The frequent k-itemset is listed for the maximum value of k and with confidence and support c and s respectively all of strong association rules are listed.

According the question

k = 3 and 3-items are {Bread, Milk, Cheese}.

Now, the rules are

Bread ∧ Cheese => Milk, [75% and 100%]

Cheese ∧ Milk => Bread, [75% and 100%]

Cheese => Milk ∧ Bread, [75% and 100%]

- b) The rule template for the brand-item-category is given below,

∀X ∈ customer, buys(X, item1) ∧ buys(X, item2) => buys(X, item3)

Listing all of the frequent k-itemset for finding the maximum k value.

Here, k= 3 and itemset = {(Wonder-bread, Dairyland-milk, Tasty-Pie),(Wonder-Bread, Sunset-Milk, Dairyland-Cheese)}

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