Measures of Central Tendency

In statistics, the central tendency refers to the location of a set of data points around the center of the data set. It represents a property that is found in all sets, although not all sets have exactly one value. The central limit theorem is an important result in probability theory which says that as numbers get larger, their distribution will tend towards normality and thus central tendency.

Multiple Choice Questions

Q.1: Magnitude of scores is included in which of the central tendency measures?

1. 1. Median
   2. Mode
   3. Mean
   4. None

Answer: (C)

Explanation: If the data set has an odd number of points, then the mean is equal to the arithmetic mean, so it also includes magnitude.

Q.2: What were the first two results of a central tendency test?

1. 1. Mean and Mode
   2. Median and Mode
   3. Mean, Median and Range
   4. None

Answer: (A)

Explanation: Mean and Mode are used to perform the test because they are the only ones that approximate a normal distribution.

Q.3: What steps are involved in a central tendency test?

1. 1. Addition, subtraction, and division.
   2. Determination of mean, median, and mode.
   3. Addition, subtraction, multiplication, and division.
   4. None of the above

Answer: (A)

Explanation: The central tendency test involves only three steps to be performed. They are addition, subtraction, and division.

Q.4: What does it mean when the central tendency is stated as the mean of samples?

1. 1. It means that the sample size is n, and the sample mean represents a set of data points taken from this sample with replacement.
   2. It means that the sample size is n, and the sample mean represents a set of data points taken from this sample with replacement.
   3. It means that the sample size is n, averaged to determine central tendency.
   4. None of the above

Answer: (A)

Explanation:  There can never be a replacement in the case of a single sample. The sample mean represents the set of data points taken from this sample with replacement.

Q.5: Which one of the following statements best describes the median measure?

1. 1. It gives us the middle score in a data set, which will not change if you rearrange the scores in any order that does not change their magnitude.
   2. It gives us the middle score in a data set, which will change if you rearrange the scores in any order that does not change their magnitude.
   3. It is always less than or equal to the mean of a data set.
   4. None of the above

Answer: (C)

Explanation: The median is the middle of a data set. If you rearrange the data, you will always get the same median.

Q.6: Which one of the following statements best describes the mode measure?

1. 1. It gives us the most common score in a data set, which will change if you rearrange the scores in any order.
   2. It gives us the most common score in a data set, which will not change if you rearrange the scores in any order.
   3. It is always greater than or equal to the mean of a data set.
   4. None of the above

Answer: (C)

Explanation: The mode will not change if you rearrange the scores in any order. This is why the rearrangement of values does not change their order insignificance.

Q.7: Which one of the following statements best describes the range measure?

1. 1. It gives us the highest and lowest scores in a data set.
   2. It gives us the lowest and highest scores in a data set.
   3. It is always greater than or equal to the mean of a data set.
   4. None of the above

Answer: (B)

Explanation: The range is the difference between the highest and lowest values in a data set. If you rearrange the data, you will always get the same range.

Q.8: Which one of the following statements best describes the standard deviation measure?

1. 1. It gives us the spread of a data set.
   2. It tells how much a data set is spread from its mean.
   3. It is always greater than or equal to the mean of a data set.
   4. None of the above

Answer: (C)

Explanation: The standard deviation is the measure of spread, so It is always greater than or equal to the mean of a data set 

Q.9: What does it mean when the central tendency was stated as the median of samples?

1. 1. It means that the sample size is n, and the sample median represents a set of data points taken from this sample with replacement.
   2. It means that the sample size is n, averaged to determine central tendency.
   3. None of the above
   4. All of the above

Answer: (A)

Explanation: Central tendency was the median of samples, which means that the sample median represents data points taken from the sample with replacement.

Q.10: If a data set is symmetrical, what does it mean?

1. 1. True distribution
   2. False
   3. True skew
   4. True symmetry

Answer: (B)

Explanation: If all the data points are, on average, the same, then it cannot be said that the skew is true or false. If all the data points are, on average, different from their mean, then it cannot be said that the skewness is true or false.

Q.11: Which one of the following statements does not contain an assumption of the standard deviation measure?

1. 1. The data points in a data set are not necessarily equal values.
   2. The distribution is normal.
   3. The data points are not all positive or negative.
   4. None of the above

Answer: (B)

Explanation: The data points are not necessarily equal values. The mean is measured with the assumption that the data set is symmetrical, which means that all of the data points can be considered the same on average. 

Q.12: What does the standard deviation measure?

1. 1. The spread of a data set.
   2. How much a data set is spread from its mean.
   3. How much a data set is spread from another data set or the mean of another data set.
   4. None of the above

Answer: (C)

Explanation: The standard deviation is based on the sample mean. It tells how much a data set is spread from its mean.

Q.13: Which one of the following statements does not contain an assumption of the standard deviation measure?

1. 1. The data points are, on average, equally spaced.
   2. The distribution is normal.
   3. The data points are not all positive or negative.
   4. All of the above

Answer: (C)

Explanation: The sample mean is calculated with the normal distribution assumption.

Q.14: Which of these statements is True?

1. 1. The sample distribution is the same as that of the population from which it came.
   2. If a data set is symmetrical, then the mean, median, and standard deviation are equal.
   3. If a data set is skewed, the mean and standard deviation are not necessarily equal.
   4. None of the above

Answer: (B)

Explanation: If all the data points in a set are equally spread (symmetric), then the mean, median, and standard deviation are equal.

Q.15: What is the standard deviation measure?

1. 1. It measures how much a data set is spread from its mean.
   2. It measures how much a data set is spread from another data set.
   3. It indicates how closely related two or more sets are in terms of their respective means, medians, and quantiles.
   4. All the above

Answer: (D)

Explanation: The standard deviation is used to find the degree of spread in a set of data values. It measures whether a sample is all about the same value or is spread out over a range of values. If all values are close, then the standard deviation will be low, but if they are spread out over a range of values, the standard deviation will be high.

Q.16: What is an assumption when a data set is normally distributed?

1. 1. The central tendency of populations is always equal to or greater than some constant.
   2. Data points in a data set are, on average, equally spaced.
   3. It has a standard deviation equal to 1.
   4. All of the above

Answer: (B)

Explanation: A normal distribution is symmetric. It has a mean equal to or greater than the median value. If all the data points are equally spread out, they must be centered on the mean, and all values will be equal on average.

Q.17: What does the standard deviation measure?

1. 1. The spread of a data set.
   2. How much a data set is spread from its mean.
   3. How much a data set is spread from another data set or the mean of another data set.
   4. None of the above

Answer: (C)

Explanation: The standard deviation is used to find the degree of spread in a set of data values. It indicates how closely related two or more sets are to each other in terms of their respective means, medians, and quantiles

Q.18: What is the Central Limit Theorem?

1. 1. A statement about the distribution of scores in samples
   2. A statement about the mean of scores from random samples
   3. A statement about the distribution of scores from a normal population
   4. A statement about the distribution of scores for any sample

Answer: (D)

Explanation: The central limit theorem states that as the size gets large, the continuous probability distribution for several scores in a set will tend towards normality and thus central tendency.

Q.19 Which of these statements is true?

1. 1. The mean is a continuous variable.
   2. The variance and standard deviation of a normal population are equal.
   3. For large samples, the distribution of scores is approximately normal.
   4. None of the above

Answer: (C)

Explanation: For large samples, the distribution will tend towards normality and thus central tendency.

Q.20 Which of the following statements is true?

1. 1. The mean is a continuous variable.
   2. The variance and standard deviation of a normal population are equal.
   3. For large samples, the distribution of scores is approximately normal.
   4. None of the above

Answer: (C)

Explanation: The standard normal distribution is the sample from which all distributions in statistics like measures of central tendency and dispersion are derived.