Univariate Data

NZ Level 5

Recognising the shape of data

Lesson

Data may be described as *symmetrical* or *asymmetrical*.

The normal distribution is the most common example of symmetrical data. The normal distribution looks like this:

The '0' point right in the middle of the distribution represents the mean, the median and the mode- all these measures of central tendency are equal. If we use that '0' line as our axis of symmetry, do you notice how the left hand side is a perfect reflection of the right hand side? That means it's symmetrical.

If your data is asymmetrical (ie. it isn't symmetrical), it may be described as *skewed*.

A positive skew means that the majority of the scores are low and the long tail is on the positive side of the peak of the graph. It is also sometimes called a right skew. This means the mean is greater than the median, which is greater than the mode.

MODE < MEDIAN < MEAN

A positively skewed graph looks something like this.

Notice how most of the scores are in the lower half of the graph?

A negative skew means that the majority of the scores are high and the long tail is on the negative side of the peak. It is also sometimes called a left skew. This means the mode is greater than the median, which is greater than the mean.

MEAN < MEDIAN < MODE

A negatively skewed graph looks something like this.

Notice how most of the scores are in the higher half of the graph?

How many peaks are there on the graph?

$\editable{}$ peaks

The table shows the number of crime novels in a bookshop for different price ranges.

Price of crime novel to nearest $\$5$$5 | Frequency |
---|---|

$5$5 | $5$5 |

$10$10 | $10$10 |

$15$15 | $17$17 |

$20$20 | $8$8 |

$25$25 | $17$17 |

$30$30 | $10$10 |

$35$35 | $5$5 |

Plot this data as a histogram.

Which of the following best describes the data in the graph and table?

Symmetrical

ARight skewed

BLeft skewed

CUniform distribution

DSymmetrical

ARight skewed

BLeft skewed

CUniform distribution

D

What type of skew is shown in the graph below?

No skew

ARight skew

BLeft skew

CNo skew

ARight skew

BLeft skew

C

Plan and conduct surveys and experiments using the statistical enquiry cycle:– determining appropriate variables and measures;– considering sources of variation;– gathering and cleaning data;– using multiple displays, and re-categorising data to find patterns, variations, relationships, and trends in multivariate data sets;– comparing sample distributions visually, using measures of centre, spread, and proportion;– presenting a report of findings