The mode is defined as the value that appears most frequently in a dataset. A dataset can be unimodal, bimodal, or multimodal if multiple values share the highest frequency of occurrence, unlike the mean or median, which are typically unique.
1142
If three values y1, y2, and y3 are identical, what is their median?
The median is the middle value of a sorted dataset. If y1 = y2 = y3, then all values are equal. Therefore, the median is equal to any of these values, including y3. This property holds for any set where all elements are identical.
1143
Which measure of central tendency is not calculated using every individual observation in the dataset?
The median is defined as the middle value of a sorted dataset. It depends primarily on the positional order of the data rather than the specific numerical values of all observations. In contrast, the mean and geometric mean require the inclusion of every data point in their respective formulas.
1144
What is the term for an index number that is calculated to represent the aggregate change across multiple items?
A simple aggregate index number is constructed by summing the prices or quantities of a group of items in a given period and dividing that sum by the corresponding sum in the base period, then multiplying by 100. It provides a composite measure for a basket of goods.
1145
Which measure of central tendency minimizes the sum of squared deviations of observations?
The arithmetic mean is the value that minimizes the sum of squared differences between itself and each observation in a dataset. Mathematically, the function f(c) = Σ(xi - c)^2 is minimized when c equals the mean. This property is fundamental in least squares estimation and regression analysis, where the mean serves as the optimal point estimate for minimizing prediction error variance.
1146
Calculate the median of the following mathematics test scores: 18, 17, 16, 15, 14, 17, 16, 14, 13, 12, 12, 11, 14, 19, 18.
To find the median, first arrange the 15 data points in ascending order: 11, 12, 12, 13, 14, 14, 14, 15, 16, 16, 17, 17, 18, 18, 19. The median is the middle value at the (n+1)/2 position, which is the 8th value. Counting to the 8th position, we find the value is 15.
1147
If a constant value of 10 is added to every observation in a dataset, how is the arithmetic mean affected?
The arithmetic mean is sensitive to changes in origin. When a constant value is added to every observation in a dataset, the mean of the new dataset will increase by exactly that same constant value. This is a fundamental property of the mean under linear transformations.
1148
Which measure of central tendency is calculated using every individual observation in a dataset?
The arithmetic mean is defined as the sum of all observations divided by the total number of observations. Because it incorporates the numerical value of every data point in the set, it is sensitive to every observation, unlike the median or mode, which are positional or frequency-based measures.
1149
What is the median of the sequence 12, 22, 32, ..., (2n + 1)2?
The sequence consists of terms of the form (k)^2 for k = 1, 2, 3, ..., 2n+1. This is a sequence of 2n+1 terms. The median of a sequence with an odd number of terms is the middle term. For 2n+1 terms, the middle position is (2n+1+1)/2 = n+1. Therefore, the (n+1)-th term in the sequence is (n+1)^2.
1150
Determine the median of the following dataset: 160, 180, 200, 280, 300, 320, 400.
The median is the middle value of an ordered dataset. Since there are 7 observations (an odd number), the median is the value at the (n+1)/2 position, which is the 4th value. Ordering the data: 160, 180, 200, 280, 300, 320, 400. The 4th value is 280.
