Tiêu đề 33 Descriptive Statistics.pdf ✅

• This set is a bi-modal observation since there are two modes • Datasets with at least three modes are called multi-modal. • If there are no repeated observations, then the set has no mode. 1.2. Grouped Data Grouped data organizes observations in a frequency distribution table. Consider this example: Score Frequency 0 – 24 2 25 – 49 5 50 – 74 26 75 - 99 18 Here, the intervals are called class intervals. The midpoint of each class interval is called the class mark. The spacing among groups is called the class width. In this example, the class width is i = 75 − 50 = 50 − 25 = 25 − 0 = 25 Measure Formula Mean x = ∑ fxm n Median x̂ = x̂lb + n 2 − cf< f i Mode x̃ = x̃lb + fm − f1 (fm − f1 ) + (fm − f2 ) i - Example: Find the example frequency table's mean, median, and mode. [SOLUTION] For the mean, Score Frequency xm fxm 0 – 24 2 12 24 25 – 49 5 37 185 50 – 74 26 62 1612 75 - 99 18 87 1566 n 51 ∑fxm 3387 x = 3387 51 x = 66.41 For the median, compute the cumulative frequency.
Score Frequency cf< 0 – 24 2 2 25 – 49 5 7 50 – 74 26 33 75 - 99 18 51 Since n 2 = 51 2 = 25.5, the median class is in the interval 50 − 74. The lower bound, therefore, is x̂lb = 49.5. x̂ = x̂lb + n 2 − cf< f i x̂ = 49.5 + 25.5 − 7 26 × 25 x̂ = 67.29 For the mode, Score Frequency 0 – 24 2 25 – 49 5 50 – 74 26 75 - 99 18 Since the maximum frequency is 26, the modal class is 50 − 74. The lower bound is x̃lb = 49.5. x̃ = x̃lb + fm − f1 (fm − f1 ) + (fm − f2 ) i x̃ = 49.5 + 26 − 5 (26 − 5) + (26 − 18) × 25 x̃ = 67.60
2. Measures of Dispersion These are also called the measures of variability. These measures describe how dispersed or how close the values of each observation are. Measure Sample Population Range Maximum – Minimum Variance s 2 = ∑(x − x) 2 n − 1 σ 2 = ∑(x − x) 2 n Standard Deviation s = √ ∑(x − x) 2 n − 1 σ = √ ∑(x − x) 2 n Mean Absolute Deviation ∑ |x − x| n Interquartile Range Q3 − Q1 Quartile Deviation Q3 − Q1 2 3. Measures of Location One may also call these quantiles. These values divide the distribution into subdivisions, and these are measures that define the division points. Most common quantiles are: quartiles, percentiles, deciles. The general formula for the quantiles is xk = xLB + kn Q − cf< f i Where: Q is the number of divisions (4 for quartiles, 10 for deciles, and 100 for percentiles) k is the division considered (e.g., k = 4 for the 4th quantile) cf< is the cumulative frequency f is the class frequency i is the class mark - Example: Find the 85th percentile of Score Frequency 0 – 24 2 25 – 49 5 50 – 74 26 75 - 99 18