Test: Statistics - 2 - UPSC MCQ

Concept use:

The relationship between mean, median, and mode in a "perfectly" symmetrical distribution is given by the empirical relationship:

Mode = 3(Median) - 2(Mean)

Calculations:

Median = (Mode + 2 × Mean) / 3

Median = (10 + 2 × 4) / 3 = 18/3 = 6

A unimodal distribution refers to a distribution that has a single peak or mode. It means that there is one value in the dataset that occurs more frequently than any other value.

When the mode is less than the mean, it indicates that the majority of the data points are located towards the higher values, while the tail extends towards the lower values. This pattern is commonly observed in negatively skewed distributions, also known as left-skewed distributions.

In a negatively skewed distribution, the tail extends towards the left side, while the bulk of the data is concentrated towards the right side. The mode represents the most frequently occurring value, and when it is lower than the mean, it suggests that the distribution is pulled towards the higher values.

To visualize this, imagine a dataset representing the test scores of a group of students. If the distribution is negatively skewed and the mode is less than the mean, it implies that there are a few students who scored exceptionally high, which extends the tail towards the left. However, the majority of the students have lower scores, resulting in the mode being lower than the mean.

Therefore, based on the given information, in a unimodal distribution where the mode is less than the mean, the distribution will most likely be negatively skewed. The tail of the distribution is longer on the left side, indicating a concentration of values towards the right side.

In a symmetrical distribution, the data points are evenly distributed around a central value, resulting in a mirror image when the distribution is folded along its center. This means that the left and right sides of the distribution are symmetrically balanced.

Given this symmetry, the mean, median, and mode will all have the same value in a perfectly symmetrical distribution.

The mean represents the average value of the dataset, calculated by summing all the values and dividing by the total number of values. The mode represents the most frequently occurring value in the dataset.

Since a symmetrical distribution has equal frequencies on both sides of the mode, the mode will be the value that occurs most often and therefore represents the highest peak in the distribution.

In a symmetrical distribution, the balance of the data on both sides of the mode implies that the mean will be the same as the mode.

Therefore, in a symmetrical distribution, the mean is equal to the mode.

In a perfectly symmetrical distribution, the data points are evenly distributed around a central value, resulting in a mirror image when the distribution is folded along its center. This means that the left and right sides of the distribution are symmetrically balanced.

Given this symmetry, the mean, median, and mode will all have the same value in a perfectly symmetrical distribution.

The mean represents the average value of the dataset, calculated by summing all the values and dividing by the total number of values. The median represents the middle value when the dataset is arranged in ascending or descending order. The mode represents the value or values that occur most frequently in the dataset.

In a perfectly symmetrical distribution, the balance of the data on both sides of the mode implies that the mean and median will be located at the center of the distribution, which is also where the mode will be located.

Therefore, in a symmetrical distribution, the mean, median, and mode are equal. They all have the same value, reflecting the central tendency and balance of the data points in the distribution.

The mean, median, and mode are three measures of central tendency that provide insights into the location or center of a dataset. While they can be equal in certain situations, it is not a universal rule that they will always be equal.

In some distributions, the mean, median, and mode may have the same value. This occurs in perfectly symmetrical distributions, such as the normal distribution, where the data points are evenly distributed around a central value.

However, in many distributions, the mean, median, and mode can have different values. This is especially true in distributions that are skewed or have multiple modes.

For example, in a positively skewed distribution, the mean will be greater than the median, and both of these may differ from the mode. Similarly, in a negatively skewed distribution, the mean will be less than the median, and the mode may be different as well.

It's important to consider the shape and characteristics of the specific distribution when determining the relationship between the mean, median, and mode. While they can be equal in some cases, it is not a guaranteed or universal outcome.

Therefore, the values of mean, median, and mode can be some times equal, but it is not always the case.

In a symmetrical distribution, the data points are evenly distributed around a central value, resulting in a mirror image when the distribution is folded along its center. This means that the left and right sides of the distribution are symmetrically balanced.

When the mean, median, and mode are equal, it indicates that the distribution is perfectly balanced and there is no skewness or bias towards either side. Each side of the distribution has an equal number of data points, resulting in a symmetrical shape.

A symmetrical distribution is also known as a normal distribution or Gaussian distribution. It follows a characteristic bell-shaped curve, where the mean, median, and mode are all located at the center of the distribution.

In summary, if the mean, median, and mode are all equal in a distribution, it indicates that the distribution is symmetrical. The balance of the data points on both sides of the distribution implies a lack of skewness or bias towards either side.

The median is a measure of central tendency that represents the middle value in a dataset when the values are arranged in ascending or descending order. It divides the dataset into two equal halves, with an equal number of values above and below it.

To find the median in an ordered array of numbers, you simply identify the value located at the center position. If the total number of values in the array is odd, there will be one middle value that is the median. If the total number of values is even, the median is typically calculated as the average of the two middle values.

For example, consider the ordered array [2, 4, 7, 9, 12]. The middle value is 7, so 7 would be the median of this array.

The median is particularly useful when dealing with skewed distributions or datasets with outliers. Unlike the mean, which is influenced by extreme values, the median provides a robust measure of central tendency that is less affected by outliers.

In summary, the middle value of an ordered array of numbers is the median. It represents the central value that divides the dataset into two equal halves when the values are arranged in ascending or descending order.

The measure of central tendency provides insights into the typical or representative value of a dataset. It aims to identify a single value that represents the central or middle part of the data.

Common measures of central tendency include the mean, median, and mode. The mean is calculated by summing all the values and dividing by the total number of values. The median represents the middle value when the dataset is arranged in ascending or descending order. The mode represents the most frequently occurring value in the dataset.

These measures help us understand where the "center" or "middle" of the dataset lies. They provide a summary or representative value that characterizes the central tendency of the group of numbers.

In contrast, measures of variability (such as the range, variance, and standard deviation) quantify the spread or dispersion of the data points. Measures of association are used to assess the relationship or correlation between two or more variables. Measures of shape describe the overall pattern or distribution of the data.

Therefore, the measure of central tendency is the one that describes the middle part of a group of numbers. It provides insights into the representative value that characterizes the central tendency of the dataset.

To find the median of the given data: 160, 180, 200, 280, 300, 320, 400, we first need to arrange the data in ascending order:

160, 180, 200, 280, 300, 320, 400

Since the total number of values in the dataset is odd (7 values), the median will be the middle value.

The middle value in this case is the fourth value, which is 280.

Therefore, the median of the given data is 280.

The mode is the value or values in a dataset that occur most frequently. In some cases, there may be multiple values with the same highest frequency, resulting in multiple modes. When this occurs, the dataset is described as having multiple modes or being multimodal.

For example, consider a dataset of exam scores: 75, 80, 85, 90, 90, 95, 95, 95. In this dataset, the value 95 occurs three times, which is the highest frequency. Therefore, the mode(s) of this dataset is 95. This dataset is said to have a mode of 95.

However, in some cases, a dataset may not have any repeated values, or all values may have the same frequency. In such cases, the dataset is considered to have no mode.

On the other hand, the mean, median, and harmonic mean are measures of central tendency that typically yield a single value. The mean is the average calculated by summing all the values and dividing by the total number of values. The median is the middle value when the dataset is arranged in ascending or descending order. The harmonic mean is a type of average used for rates or ratios.

Therefore, the measure of average that can have more than one value is the mode.

The geometric mean is a measure of central tendency that is commonly used for a set of positive numbers. It is calculated by taking the nth root of the product of n positive values.

Since the geometric mean involves taking the root of positive values, it cannot be negative. This is because taking the root of a negative number or zero is not defined in standard mathematical operations.

On the other hand, the median, arithmetic mean, and harmonic mean can be negative under certain circumstances. For example, if a dataset contains negative values, the median and arithmetic mean can be negative if the negative values outweigh the positive values.

Therefore, among the options given, the measure that cannot be less than zero (negative) is the geometric mean. It is specifically designed for positive values and does not yield negative results.

The harmonic mean is specifically designed for rates or ratios, making it suitable for calculating average speeds. It is calculated by taking the reciprocal of each value, finding their arithmetic mean, and then taking the reciprocal of that result.

When finding the average speed of a journey, it is common to have different segments or intervals with varying speeds. The harmonic mean is useful in this scenario because it gives more weight to the slower speeds.

The harmonic mean ensures that the calculated average reflects the overall time taken for the journey, considering the different speeds and distances traveled in each segment. By taking the reciprocal of the speeds, finding their arithmetic mean, and then taking the reciprocal again, the harmonic mean effectively balances the impact of different speeds on the average.

On the other hand, the mean, geometric mean, and weighted mean are not as appropriate for finding the average speed of a journey. The mean does not account for the different speeds and distances traveled. The geometric mean is more suitable for multiplicative relationships rather than additive ones like average speeds. The weighted mean involves assigning different weights to each value, which may not be necessary unless there are specific considerations for certain segments of the journey.

Therefore, to find the average speed of a journey, the appropriate measure of central tendency is the harmonic mean. It accounts for the varying speeds and ensures the average reflects the overall time taken for the journey.

The harmonic mean is a measure of central tendency that is calculated by taking the reciprocal of each value, finding their arithmetic mean, and then taking the reciprocal of that result.

When computing the harmonic mean, taking the reciprocal of a value means dividing 1 by that value. However, division by zero is undefined in mathematics. Therefore, if any value in the data set is zero, it becomes impossible to compute the harmonic mean because it would involve division by zero.

On the other hand, the mode, median, and mean can still be computed even if there is a zero value in the data set.

• The mode represents the most frequently occurring value, which can still be determined even if there is a zero value present.

• The median is the middle value when the data set is arranged in ascending or descending order. If there is a zero value, it will still have a position in the order, and the median can be determined accordingly.

• The mean can be computed by summing all the values and dividing by the total number of values. The presence of a zero value does not prevent the mean from being calculated.

Therefore, if any value in the data set is zero, it is not possible to compute the harmonic mean, but it is still possible to compute the mode, median, and mean.

The geometric mean is a measure of central tendency that is calculated by taking the nth root of the product of n positive values. It is commonly used when dealing with multiplicative relationships or ratios.

While the geometric mean is typically used with positive values, it can still be computed with negative values present in the data set. However, it is important to note that the presence of negative values can affect the interpretation and usefulness of the geometric mean.

Negative values in the data set will result in complex or imaginary numbers when calculating the nth root of the product. This means that the geometric mean may not always have a meaningful interpretation when negative values are involved.

On the other hand, the arithmetic mean, harmonic mean, and mode can still be computed even if there are negative values in the data set.

• The arithmetic mean, or simply the mean, is calculated by summing all the values and dividing by the total number of values. The presence of negative values does not prevent the mean from being calculated.

• The harmonic mean is calculated by taking the reciprocal of each value, finding their arithmetic mean, and then taking the reciprocal of that result. The presence of negative values does not prevent the harmonic mean from being computed.

• The mode represents the most frequently occurring value, which can still be determined even if there are negative values present.

Therefore, if any value in the data set is negative, it is not impossible to compute the arithmetic mean, harmonic mean, and mode. However, it is important to consider the implications and limitations when negative values are involved. The geometric mean may still be computed, but the interpretation may not be meaningful in such cases.

The median is a measure of central tendency that represents the middle value of a dataset. To find the median, the data points need to be arranged in order from least to greatest (ascending order) or from greatest to least (descending order).

By arranging the data in this manner, it becomes easier to identify the middle value. If the dataset contains an odd number of values, the middle value is the one that falls exactly in the center. If the dataset contains an even number of values, the median is typically calculated as the average of the two middle values.

For example, let's consider the following dataset: 5, 10, 3, 8, 2, 9. To find the median, we need to arrange the data in ascending or descending order:

Ascending order: 2, 3, 5, 8, 9, 10 Descending order: 10, 9, 8, 5, 3, 2

By arranging the data in either ascending or descending order, we can easily identify the middle value or values to calculate the median.

In summary, if someone wants to compute the median, it is necessary to arrange the data either in ascending or descending order. This allows for the determination of the middle value or values, which represent the median of the dataset.

The arithmetic mean, also known as the mean, is calculated by summing all the values in the data set and dividing by the total number of values. It represents the balance point or center of the data.

Extreme values in the data set can have a significant impact on the arithmetic mean because they contribute to the overall sum. If there are extreme values that are very small or very large, they can pull the mean towards those extreme values.

For example, consider the following data set: 1, 2, 3, 4, 1000. The arithmetic mean of this data set is (1 + 2 + 3 + 4 + 1000) / 5 = 202. If we remove the extreme value of 1000, the mean becomes (1 + 2 + 3 + 4) / 4 = 2.5. The presence of the extreme value significantly affects the arithmetic mean.

On the other hand, the geometric mean, median, and harmonic mean are less influenced by extreme values.

• The geometric mean is calculated by taking the nth root of the product of n values. Since extreme values contribute to the product rather than the sum, their effect is mitigated.

• The median represents the middle value when the data set is arranged in ascending or descending order. Extreme values do not impact the position of the middle value, making the median less affected by them.

• The harmonic mean is calculated by taking the reciprocal of each value, finding their arithmetic mean, and then taking the reciprocal of that result. Extreme values have a smaller influence on the harmonic mean due to the reciprocal operations involved.

In summary, the measure of average that is affected by extreme values in the data set is the arithmetic mean. Extreme values can significantly alter the mean due to their contribution to the overall sum.

The mode is the value or values that occur most frequently in the data set. It represents the most common observation(s) or the peak of the distribution.

Unlike the arithmetic mean, geometric mean, and median, the mode does not take into account all the values in the data set. Instead, it focuses solely on identifying the value(s) with the highest frequency.

For example, consider the following data set: 2, 4, 4, 6, 6, 6, 8, 8, 8. In this case, the mode is 6 because it occurs three times, which is more frequently than any other value. The mode is determined by counting the occurrences of each value, rather than considering the entire range of values.

On the other hand:

• The arithmetic mean is calculated by summing all the values in the data set and dividing by the total number of values. It incorporates all the values in the calculation.

• The geometric mean is calculated by taking the nth root of the product of n values. It also considers all the values in the data set.

• The median represents the middle value when the data set is arranged in ascending or descending order. It includes all the values and identifies the middle observation(s).

Therefore, among the options given, the measure of average that is not based on all the values given in the data set is the mode. It focuses on identifying the most frequently occurring value(s) rather than considering all the values in the data set.

The mode is a measure of central tendency that represents the value or values in a data set that occur most frequently. It is the observation(s) with the highest frequency.

In other words, the mode represents the most popular or commonly occurring value in the data set. It is the value that appears more often than any other value.

For example, consider the following data set: 3, 5, 5, 7, 7, 7, 9, 9, 9. In this case, the mode is 7 and 9 because they both occur three times, which is the highest frequency. Both 7 and 9 are the most repeated values in the data set.

The mode is particularly useful when you want to identify the value(s) that have the highest occurrence or when you are interested in the most typical observation in the data set.

On the other hand:

• The median represents the middle value when the data set is arranged in ascending or descending order.

• The mean, also known as the arithmetic mean, is calculated by summing all the values and dividing by the total number of values.

• The geometric mean is calculated by taking the nth root of the product of n positive values.

Therefore, among the options given, the most repeated (popular) value in a data set is called the mode. It represents the value(s) that occur with the highest frequency in the data set.

The geometric mean is calculated by taking the nth root of the product of n positive values. However, it is important to note that the geometric mean is only defined for positive values. It cannot be calculated when negative values or zero are present in the dataset.

In the given values -2, 4, 03, 6, 0, we have a negative value (-2) and a zero (0). Since the geometric mean cannot be computed with negative values or zero, we cannot find the geometric mean for this dataset.

Therefore, the geometric mean of -2, 4, 03, 6, 0 is cannot be computed.

Both the mean and variance are statistical measures that provide insights into different aspects of a dataset.

The mean, also known as the arithmetic mean or average, is calculated by summing all the values in the dataset and dividing by the total number of values. It represents the central tendency or average value of the dataset. To obtain an accurate mean, all values in the dataset are considered and included in the calculation.

The variance is a measure of the dispersion or spread of the dataset. It quantifies the average squared deviation from the mean. To calculate the variance, each value in the dataset is subtracted from the mean, squared, and then summed. Again, all values in the dataset are taken into account in the variance calculation.

Both the mean and variance require consideration of all values in the dataset to provide meaningful and accurate results. Excluding any values would lead to an incomplete representation of the data and could potentially introduce biases or inaccuracies in the calculations.

Therefore, the calculation of mean and variance is based on all values in the dataset. It is important to include all values to obtain reliable and comprehensive measures of central tendency and dispersion.