Question

Review the given data with the following statistical measures:

Mode = 70
Mean = 54
Median = 60
1st Quartile = 28
3rd Quartile = 71
a) Define each statistical measure (mode, mean, median, 1st quartile, 3rd quartile) and provide a brief explanation of its significance in describing a data set.
b) Discuss the implications of having a mode, mean, and median that are not equal, considering the shape and symmetry of the data distribution.
c) Analyze the interquartile range, represented by the 1st and 3rd quartiles, and discuss how it provides insights into the spread or variability of the data.

Ensure a thorough review and interpretation of the given data based on the provided statistical measures.

Answer 1

The statistical measures discussed include the mode (most frequent value), mean (average), median (middle value), first quartile (25th percentile), and third quartile (75th percentile). The non-equality of mode, mean, and median suggests data skewness, and the interquartile range indicates variability in the data.

Step-by-step explanation:

Understanding Statistical Measures

To address the given data with the described statistical measures, we'll define each measure and discuss their implications in analyzing the data set. The mode is the most frequently occurring value in a data set. In this case, the mode is 70. The mean is the average of all the values, which is indicated as 54. The median is the middle value when the data is ordered, and it's given as 60.

The first quartile (Q1) is the median of the lower half of the data, excluding the median of the entire data set if applicable, represented here as 28. It means 25% of the data is less than or equal to this value. The third quartile (Q3) is similar but for the upper half, and it's 71, meaning 75% of the data is less than or equal to this value.

When the mode, mean, and median are not equal, it suggests the data set is not symmetrical and is likely skewed. In this instance, the mode is higher than both the mean and median, indicating a skew to the right. The large difference between the first quartile and the mean suggests that there are lower outliers or a lot of low values that are dragging the mean down.

The interquartile range (IQR) is calculated by subtracting Q1 from Q3, which measures the middle 50% spread of the data. A larger IQR represents greater spread or variability. In this data set, the IQR would be Q3 - Q1 = 71 - 28 = 43, indicating that there's a considerable variation between the middle 50% of the values.

Answer 2

The statistical measures of mode, mean, median, 1st quartile, and 3rd quartile offer different perspectives on the central tendency and spread of a data set. Discrepancies among these measures indicate a non-symmetrical distribution. The interquartile range provides a measure of data variability and helps in identifying outliers.

Step-by-step explanation:

Understanding Statistical Measures

The given data presents various statistical measures. Here is the definition and significance of each:

• Mode: The most frequently occurring value in a data set. Having a mode of 70 suggests that this value appears more often than others.
• Mean: The average of the data set, calculated by adding all values and dividing by their count. A mean of 54 implies the central point of data values.
• Median: The middle value when a data set is ordered from least to greatest. A median of 60 indicates that half of the values are above and half below this point.
• 1st Quartile (Q1): Also known as the 25th percentile, it marks the value below which 25% of the data falls. A 1st quartile of 28 implies 25% of data values are less than this.
• 3rd Quartile (Q3): The 75th percentile, below which 75% of the data falls. The 3rd quartile at 71 indicates that most of the data is at or below this value.

When the mode, mean, and median are not equal, it suggests the data is not symmetrical and may be skewed. The discrepancy between these values indicates the presence of outliers or a skewed distribution.

The interquartile range (IQR), calculated as Q3 minus Q1 (71 - 28 = 43), represents the spread of the middle 50% of the data. It offers insights into the data's variability and helps identify outliers.