Statistics 1 Dersi 2. Ünite Sorularla Öğrenelim

Define central tendency. 

Central tendency is defined as “the tendency of data to cluster around some random variable value”.

What is the most frequent value in a set of numbers called?

The mode is the most frequent value in the entire data.

What is the middle value of an ordered dataset called?

The median is the middle value of an ordered dataset.

What are the purposes of quantifying central tendency measures?

There are several purposes for quantifying central tendency measures. The primary purpose of computing them is to determine a single value which may be used to indicate the center of an entire data set including magnitudes of the same data. Another purpose is that since measure of center may represent the whole data, it enables us to make comparisons within or between groups of data. For example, average test performance of students in a class can be compared with the average test performance of students in other classes at the same grade level. Third purpose is that average value can be used in computing some other statistical measures like skewness, kurtosis, and dispersion. More technically, there is an idea that it would be more accurate to calculate different measures of location for different situations, considering the distribution of data.

17 10 13 11 12 7 10 9 10 16 17 10

What is the mode of the given data set?

The answer is 10  because it is the value that occurs most often than any other observation in the data (it is repeated 4 times).

At home, there are 10 guests and here are their ages: 75, 55, 12, 16, 30, 42, 9, 23, 61, 37. 

What is the arithmetic mean of this population?  

(75 + 55 + 12 + 16 + 30 + 42+ 9 + 23 + 61 + 37)/10= 36

The answer is 36. 

Please, write three properties of arithmetic mean.

• The mean is accepted as the representative measure of location for the entire data since it is based upon all values.
• It is easily understood and calculated.
• Several means can also be comparable when they are measured on the same scale.
• The mean cannot be measured for nominal or ordinal data. Although mean can be calculated for numerical ordinal data (e.g., survey on Likert type scales), many times it does not give a meaningful information about the data.
• The mean is a valid measure of central tendency when we have scale type data such as interval or ratio data and the distribution is normal.
• Unlike the mode and the median, the mean is capable of further algebraic operations due the type of data.
• The mean is very sensitive to extreme low and extreme high scores indicating skewed distributions.

What is an arithmetic mean of the extremes in both end of the data set called?

Midrange is an arithmetic mean of the extremes in both end of the data set.

What is the formula for calculating midrange?

Midrange = X(max) + X(min)/ 2

Suppose that smart phone prices range from ?1000 (the cheapest) to ?7000 (the most expensive) in a store. Calculate the midrange of the phones.  

Midrange= (X(max) + X(min)) / 2

(?1000 + ?7000)/2= 4000 

What are the most widely used central tendency measures?

The arithmetic mean, median, and mode are the most widely used central tendency measures.

Which one is the simplest but the weakest meausre of center: mode, median or mean? Why?

The mode is the simplest but it is the weakest measure of center. If all values have the same frequency, there is no mode in the data or if two or more values occur with the same frequency, there is multiple modes. In these cases, the mode provides limited information about the center of data. The median is a more useful measure since it represents a more typical score and many people can easily understand it. The mean is a more valid measure of center because it takes into account all the values in a dataset. For this reason, the mean is the most powerful and preferred measure of central tendency in many disciplines.

Under what condition is it NOT possible to find the mode? 

When all the values have the same frequency, we have NO mode.  

Here are the exam results of 9 students taking the same course:

65, 45, 65, 60, 75, 65, 55, 50, 50

What is the mode of this population? 

45 (1)

50 (2)

55 (1)

60 (1)

65 (3)

75 (1)

The mode is 65 because it is the value that occurs most often than any other observation in the data set.

Here are the exam results of 9 students taking the same course:

65, 45, 65, 60, 75, 65, 55, 50, 50

What is the median of this population? 

The median of the data set is the value that will be in the middle of the data set when it is ordered from smallest to largest. So median is 60.

45, 50, 50, 55, 60, 65, 65, 65, 75

Here are the exam results of 9 students taking the same course:

65, 45, 65, 60, 75, 65, 55, 50, 50

What is the arithmetic mean of this population? 

(65 + 45 + 65 + 60 + 75 + 65 + 55 + 50 + 50)/9 = 53,33

What is the average of a set of numbers in the data called?

The mean refers to the average of a set of numbers in the data.

Which one is a less resistant central tendency: mode, median or mean? Why? 

The mode and median are the more resistant central tendency measures since their calculation does not require the use of all the values in the data. On the other hand, the mean (e.g., arithmetic, geometric, and weighted mean) is a less resistant measure of location, these methods require use of all the values, including outliers, in the data. So the answer is mean.

Define crude mode.

In a grouped frequency distribution, the midpoint of the class with the highest frequency is an estimate of the mode. This mode is sometimes called as the crude mode.

Which one is the main centrality measure for nominal scales: mode, median or mean?

The mode is the main centrality measure for nominal scales (categorical) whereas the mean and the median are not meaningful measures for nominal variables.