Mongolian provinces are divided into 5 regions, researches were made by sample survey for children of Bayan Ulgii province from Western region, children of Dornod province from Eastern region, children of Umnu Gobi province from Central region, children of Khuvsgul province from Khangai region and children of Sukhbaatar District from Ulaanbaatar city.

Policy of Mongolian regional development [5] (Figure 2).

3 - 5 years old children of selected provinces and city who participated in the research (Table 1).

The datas were collected by mail and online from teachers of kindergardens from selected regions. Due to COVID-2019, all education system has transferred to online.

The research has started on the 7 January 2021, Mathematics was selected from the preschool education curriculum following “Music and Fine Arts”, “Mathematics”, “Nature and Social Environment”, “Language”, “Movement and Health” and “Development of Socializing”.

Research data were based on performance of the children according to 5 tasks of the Mathematics [6]. Following are:

Task 1: sorting one and plural;

Task 2: sorting items by size and naming;

Task 3: sorting items by similarity, nominating and connecting;

Task 4: using and recognizing position words;

Task 5: using and recognizing time expressions: now and after.

Task evaluation: it was evaluated such as “very good-5 scores”, “good-4 scores”, “satisfied-3 score”, “bad-3 score”, “very bad-3 score”. The survey wasn’t done due to corona, schools and kindergartens were locked down. Each child performance of the tasks 1 - 5. Tasks 1 - 5 are indicated such as: q1, q2, q3, q4, q5 (Table 2).

Analysis of the Data mining of the Hierarchical Clustering on Principal Components—HCPC is possible to calculate by algorithms of methods PCA, MCA, MFA in program R [4] [7]. Evaluation mathematical ability of the children is measured by next four formulas.

There are as follows, where |p − p'| is the distance between two objects or points, p and p'; m_i are the mean for cluster, C_i; and n_i is the number of objects in C_i.

Minimum distance:

d i s t ( c i , c j ) min p ∈ c i , p ′ ∈ c j { | p − p ′ | } min (3.1)

Maximum distance:

d i s t ( c i , c j ) max p ∈ c i , p ′ ∈ c j { | p − p ′ | } max (3.2)

Mean distance:

d i s t mean ( c i , c j ) = | m i − m j | (3.3)

Average distance:

d i s t min ( C i , C j ) = 1 n i n j ∑ p ∈ c i , p ∈ c j | p − p ′ | (3.4)

Hierarchical Clustering on Principal Components

The formula was made classification of each cluster depending distance to the nearest cluster center.

max i ∈ q min q ′ ≠ q d ( i , C q ′ ) (3.5)

C_q’ to cluster center q' distance, it is classified maximum and minimum mean of the nearest and farthest distance and different means.

Using this formula and calculation, it is described difference of it result of cluster analysis on age classification for children according to the regional zone.

Cluster analysis were made on database at 3 years old participants of kindergarden from selected provinces and urban zone according to the Mongolian regional zone. Formula (3.3). Three divisions of the Cluster analyses on the Hierarchical Clustering on Principal Components. It was used fviz_dend() function of the method HCPC [7] (Figure 3). According to regional zone, data of 3 years old participants at age of 3 - 5 years old were included in 3 clusters by cluster analyses.

In (Figure 3) 3 divisions of the cluster analysis involving total 3 years old participants. Hierarchical Clustering on Principal Components are divided into 3 clusters, it is described different means that the nearest and farthest point of the central point. Formulas (3.1)-(3.5).

The result of the analysis of the Hierarchical Clustering on Principal Components, using algorithms (MFA) of methods HCPC, it is described different means that the nearest and farthest point of the each cluster (Figure 4).

Total 3 years old children of the cluster task performance are 76.6%, distance to the cluster center is 8.1%.

Here (Figure 5). It is 0.25 mean that green the nearest distance to the cluster center, 0.50 mean is blue middle and 0.75 mean is red far [8]. Algorithm (PCA) of the method HCPC.

The task performance 1 - 5 for the total 3 years old participants of each cluster is 76.58%, result of each cluster.